American Mathematical Monthly Geometry Problems 1894 –

YIU : Problems in Elementary Geometry 1

Vis171. (Marcus Baker) In a traingle ABC, the center of the circumscribed circle is O, the center of the inscribed circle is I, and the orthocenter is H. Knowing the sides of the triangle OIH, determine the sides of triangle ABC. R − 2 Solution by W.P. Casey: Let N be the nine-point center. From IN = 2 r and OI = R(R − 2r), R and r can be determined. The circumcircle and the incircle, and the nine-point circle, can all be constructed. Then, Casey wrote, “[i]t only remains to find a point C in the circumcircle of ABC so that the tangent CA CA to circle I may be bisected by the circle N in the point S, which is easily done”. [sic] The construction of ABC from OIH cannot be effected by ruler and compass in general. Casey continued to derive the cubic equation with roots cos α, cos β,cosγ, and coefficients in terms of R, r,and := OH,namely, r 4(R + r)2 − (2 +3R2) R2 − 2 x3 − (1 + )x2 + x − =0. R 8R2 8R2 Given triangle OIH,letN be the midpoint of OH. Construct the circle through N tangent to OI at O.ExtendIN to intersect this circle again at M. The diameter of the circumcircle is equal to the length of IM. From this, the circumcircle, the nine-point circle, and the incircle can be constructed. Now, it remains to select a point X on the nine point circle so that the perpendicular to OX is tangent to the incircle. This is in general not constructible.

√ √ √ √ Here is an example: Let OI =2 2α, OH = 42α,andIH = 15α,whereα = 17. Then s − a, s − b, s − c are the roots of the equation x3 − 34x2 +172x − 2 · 172 =0. YIU : Problems in Elementary Geometry 2

The roots of this equation are inconstructible.

Vis186. (Marcus Baker) (Malfatti circles) Inscribe in any plane triangle three circles each tangent to two sides and the other two circles. Three solutions were published. The second one, by U. Jesse Knisely, contains a construction:

Thethirdsolution,byE.B.Seitz,1 gives the radii of the Malfatti circles as

(1 + tan β )(1 + tan γ ) 4 4 · r x = α , 1+tan 4 2 and analogous expressions for the other two.

Vis52. (James McLaughlin) (Inverse Malfatti problem) Three circles, radii p, q, r,are drawn in a triangle, each circle touching the other two and two sides of the triangle. Find the sides of the triangle. Solution by E.B.Seitz: write σ = p + q + r. √ √ √ √ √ √ √ √ √ pσ +(p( p + q + r) − pqr) σ − pqr( q + r − p) a = √ √ √ √ √ √ √ √ (q + r)2. (p q + q p + pqσ − r σ)(p r + r p + prσ − q σ)

Vis289. (Christine Ladd) If R is the radius of the circumscribed circle of a triangle ABC, r the radius of the inscribed circle, p the radius of the circle inscribed in the orthic triangle, I the center of the inscribed circle of the triangle ABC,andQ the center of the circle inscribed 1The very ﬁrst issue of the Monthly contains a biography of Seitz, written by Finkel. YIU : Problems in Elementary Geometry 3 in the triangle formed by joining the midpoints of the sides of ABC, show that 1 QI2 = . 2(7Rp − 6Rr +3r2 +2R2)

This certainly is not right! The correct formula should be 1 QI2 = (2Rp − 6Rr +3r2 +2R2). 2

Q is the Spieker center X10 and 1 QI2 = (−a3 +2a2(b + c) − 9abc). 8s

IG 2 Also, IX10 = 3 .Now, − 3 2 − − 3 − 2 − 2 ( a +2a (b + c) 9abc)= s1 +5s1s2 18s3 =2s(5r 16Rr + s ).

2 1 2 − 2 Therefore, QI = 4 (5r 16Rr + s ).

YZ Now, for the orthic triangle, sin α =2R cos α. This means that YZ = R sin 2α.Thesemi- perimeter of the orthic triangle is therefore 1 abc 4·R s = R(sin 2α +sin2β +sin2γ)=2R sin α sin β sin γ = = = . 2 4R2 4R2 R Also, =2cosα cos β cos γ. YIU : Problems in Elementary Geometry 4

The inradius of the orthic triangle is p = =2R cos α cos β cos γ. s Now, (b2 + c2 − a2)(c2 + a2 − b2)(a2 + b2 − c2) cos α cos β cos γ = 8a2b2c2 − 6 4 − 2 2 − 3 − 2 ( s1 +6s1s2 8s1s2 8s1s3 +16s1s2s3 8s3 = 2 8s3 32r2s2[s2 − (2R + r)2] = 8(4Rrs)2 s2 − (2R + r)2 = . 8R2 s2−(2R+r)2 Therefore, p = 4R .

s2 =(2R + r)2 +4Rp. Finally,

1 QI2 = (5r2 − 16Rr + s2) 4 1 = [5r2 − 16Rr +(2R + r)2 +4Rp] 4 1 = [6r2 − 12Rr +4R2 +4Rp] 4 1 = (2Rp − 6Rr +3r2 +2R2). 2

VIS 274. If the midpoints of three sides of a triangle are joined with the opposite vertices and R1,...,R6 are the radii of the circles circumscribed about the 6 triangles so formed, and r1, r2,...,r6 the radii of the circles inscribed in these triangles. prove that R1R3R5 = R2R4R6 and 1 1 1 1 1 1 + + = + + . r1 r3 r5 r2 r4 r6

VIS 282 (W.P.Casey) Let X, Y , Z be the traces of the incenter. Construct circles through I and with centers at X, Y , Z. The sum of the reciprocals of the radii of all the circles touching these circles is equal to four times the reciprocal of the inradius. YIU : Problems in Elementary Geometry 5

G1.941.S941. (B.F.Finkel) Show that the bisectors of the angles formed by producing the sides of an inscribed quadrilateral intersect each other at right angles.

G2.941.S942;943. (From Bowser’s Trigonometry) Show that π 2 · 4 · 6 · 8 · 10 ··· 2 = , 2 1 · 3 · 5 · 7 · 9 ··· Wallis’ expression for π. P.H. Philbrick pointed out that this expression was not correct. It should be π 2 · 4 · 6 · 8 · 10 ···(2n) 2 1 = · . 2 1 · 3 · 5 · 7 · 9 ···(2n − 1) 2n +1

G3.941. (From Todhunter’s Trigonometry)IfA be the area of the circle inscribed in a triangle, A1, A2, A3 the ares of the escribed circles, show that 1 1 1 1 √ = √ + √ + √ . A A1 A2 A3

G4.941.S943. (From Todhunter’s Trigonometry) Three circles whose radii are a, b,andc touch each other externally; prove that the tangents at the points of contact meet in a point whose distance from any one of them is abc . a + b + c YIU : Problems in Elementary Geometry 6

G5.941.S943. (Adolf Bailoﬀ) If from a variable point in the base of an isosceles triangle, perpendiculars are drawn to the sides, the sum of the perpendiculars is constant and equal to the perpendicular let fall from either extremity of the base to the opposite side.

G6.941.S94:159–160. (Earl D. West) Having given the sides 6, 4, 5, and 3 respectively of a trapezium, inscribe in a circle, to ﬁnd the diameter of the circle. [Note: trapezium means quadrilateral].

G7.941.S94:160. (William Hoover) Through each point of the straight line x = my + h is drawn a chord of the parabola y2 =4ax, which is bisected in the point. Prove that this chord touches the parabola (y − 2mn)2 =8a(x − h).

G8.941.S944. (Adolf Bailoﬀ) If the two exterior angles at the base of a triangle are equal, the triangle is isosceles.

G9.941.S94:160–161. (J.C.Gregg) Two circles intersect in A and B. Through A two lines CAE and DAF are drawn, each passing through a center and terminated by the circumferences. Show that CA · AE = DA · AF . (Euclid)

G10.941.S94:161. (Eric Doolittle) If MN be any plane, and A and B any point without the plane, to ﬁnd a point P , in the plane, such that AP + PB shall be a minimum.

G11.941. (Lecta Miller) A gentleman’s residence is at the center of his circular farm containing a = 900 acres. He gives to each of his m = 7 children an equal circular farm as large as can be made within the original farm; and he retains as large a circular farm of which his residence is the center, as can be made after the distribution. Required the area of the farms made. YIU : Problems in Elementary Geometry 7

G12.S94:198–199. (J.F.W.Scheﬀer) Let OA and OB represent two variable conjugate semi - diameters of the ellipse x2 y2 + =1. a2 b2 On the chord AB as a side describe an equilateral triangle ABC. Find the locus of C.

G13.941. (Henry Heaton) Through two given points to pass four spherical surfaces tangent to two given spheres.

G14.S94:232–233;268–269. (Henry Heaton) Through a given point to draw four circles tangent to two given circles.

G15.S94:233–234. (Issac L. Beverage) A man starts from the center of a circular 10 acre ﬁeld and walks due north a certain distance, then turns and walks south - west till he comes to the circumference, walking altogether 40 rods. How far did he walk before making the turn?

G16.941. (H.C.Whitaker) Three lights of intensities 2, 4, and 5 are placed respectively at points the coordinates of which are (0, 3), (4, 5) and (9, 0). Find a point in the plane of the lights equally illuminated by all of them.

G17.942:S94:269. (Robert J. Aley) Draw a circle bisecting the circumference of three given circles.

G18.942.S94:271. (Henry Heaton) Through two given points to draw two circles tangnet to a given circle.

G19.942:S94:315. (A. Calderhead) If any point be taken in the circumference of a circle, and lines be drawn from it to the three angles of an inscribed equilateral triangle, prove that the middle line so drawn is equal to the sum of the other two.

G20.942. (George Bruce Halsted) Demonstrate by pure spherical geometry that spherical tangents from any point in the produced spherical chord common to two intersecting circles on a sphere are equal.

G21.943. (Charles E. Myers) A cistern 6 feet in diameter contains 3 inches of water. If a cylinder, four feet long and one foot in diameter, be laid in a horizontal position on the bottom, to what height will the water rise?

G22.943.S94:316–317. (J.A.Timmons) Given the perimeter of a triangle = 100(2s), the YIU : Problems in Elementary Geometry 8 radius of the inscribed circle = 9(r), and the radius of circumscribed circle = 20(R); it is required to ﬁnd (1) the sides of the triangle, (2) the radius of the circle circumscribing the triangle formed by bisecting the exterior angles of the original triangle, (3) the area of the triangle thus formed, all in terms of R, r,ands.

G23.943.S94:352–353. (E.L.Platt) The ordinate of the point P of an ellipse is produced to meet the circle described on the major axis as diameter at Q. CQ, the straight line joining Q and the center of the ellipse, is tangent to the circle described on the focal radius of P as diameter. If θ is the excentric angle of P ,provethat 2(2a + b) ± 4 a(a + b) sin 2θ = . a − b

G24.943.S94:353–354. (T.W.Palmer) Two right triangles have the same base, the hypotenuse of the ﬁrst is equal to 60, of the second 40. The point of intersection of the two hypotenuse is at the distance 15 from the base. Find the length of the base.

G25.943.S94:354–355;434. (L.B.Fraker) The sides of a quadrilateral board are AB =7, BC = 15, CD = 21, and DA = 13; radius of inscribed circle is 6. (1) What are the dimensions of the largest rectangular board that can be cut out of the given board, (2) largest square, (3) largest equilateral triangle?

G26.943.S94:355, (J.F.W.Scheffer) ABCD represents a rectangle, and ABEF atrape- zoid which is perpendicular to the rectangle, both figure having the side AB common to eahc other, and ADF and BCE forming two right triangles perpendicular to the rectangle ABCD. To determine the conoidal surface CDFE so as to satisfy the condition tha any plane laid through AB will intersect it in a straight line. Also find volume of the solid thus formed.

G27.943.S94:355–356. (Adolf Bailoﬀ) A line BE, that bisects an angle exterior to the vertical angle of an isosceles triangle is parallel to the base AC.

G28.944. (Henry Heaton) Through three given points to pass two spherical surfaces tangent to a given sphere.

G29.944:S94:395. (H.W. Holycross) If the two angles at the base of a triangle are bisected; and through the point of meeting of the bisectors a line is drawn parallel to the base, the length of the parallel between the sides is equal to the sum of the segments of the sides between the parallel and the base. YIU : Problems in Elementary Geometry 9

G30.944.S94:395–396. (Charles E. Myers) A circle containing one acre is cut by another whose center is on the cirucmference of the given circle, and the area common to both is one - half acre. Find the radius of the cutting circle.

G32.94:162.S95:16–17. (W. Hoover) If a conic be inscribed in a triangle and its focus moves along a given straight line, the locus of the other focus is a conic circumscribing the triangle. Proof (Hoover). The product of the perpendicular distances from the foci of a conic to a tangent is constant. Use trilinear coordinates.

G33.94:317.S95:17. (B.F. Sine) If a given circle is cut by another circle passing through two ﬁxed points the common chord passes through a ﬁxed point.

− R G40:S95:156. (J.C.Corbin) r1 + r2 + r3 r = 4 . This can be found in Chauvenet’s Geometry.

G42:S95:157–158;189–191. Steiner-Lehmus Theorem (6 proofs).

G43.S95:80. (J.F.W.Scheﬀer) The consecutive sides of a quadrilateral are a, b, c, d. Supposing its diagonals to be equal, ﬁnd them and also the area of the quadrilateral.

G45.95:122.S95:274–276. (B.F.Burleson) Determine the radius of a circle circumscribing three tangent circles of radii a = 15, b =17andc = 19. Descartes’ formula; answer not particularly elegant.

G46.95:158.S95:318–319. (G.E. Brockway) If an equilateral triangle is inscribed in a circle, the sum of the squares of the lines joining any point in the circumference to the three vertices of the triangle is constant.

G47.95:158.S95:319-320. (J.C. Gregg) Given two points A and B and a circle whose center is O, show that the rectangle contained by OB and the perpendicular from B on the polar of A is equal to the rectangle contained by OB and the perpendicular from A on the polar of B.

G48.95:233 (I.J.Schwatt) The Simson line belonging to one point of intersection of Bro- card’s diameter of a triangle with the circumcircle of this triangle is either parallel or perpendicular to the bisector of the angle formed by the side BC of triangel ABC and the corresponding side BC of Brocard’s triangle. YIU : Problems in Elementary Geometry 10

G49.95:233,S96:56–57. (J.C. Williams) Of all triangles inscribed in a given segment of a circle, with the chord as base, the isosceles is the maximum.

G50.95:276. (B.F.Finkel) Draw a line perpendicular to the base of a triangle dividing the triangle in the ratio of m : n.

G50:95.276. (G.B.M.Zerr) To construct a trapezoid; given the bases, the perpendicular distance between the bases and the angle formed by the diagonals.

G53.S95.320. (B.F.Finkel) A pole, a certain length of whose top is painted white, is standing on the side of a hill. A person at A observes that the white part of the pole subtends an angle equal to α andonwalkingtoB,adistanced, directly down the hill towards the foot of the pole the white part subtends the same angle. What is the length of the white part, if the point B is at a distance b from the foot of the pole?

G54.95:57. (I.J. Schwatt)2 If through the center of perspective D of a given triangle ABC and its Brocard triangle A B C be drawn straight lines so as to pass through Sa, Sb, Sc (the midpoints of the sides BC, CA, AB)andifSaD1 is made equal to DSa, SbD2 equal to DSb, and ScD3 equal to DSc then are (1) the ﬁgures D1O AO, D2O BO, D3O CO parallelograms (O and O are Brocard’s points), (2) the triangles D1D2D3 and ABC are equal, and (3) D1A, D2, D3C intersect in S (the midpoint of OO ).

2University of Pennsylvania. YIU : Problems in Elementary Geometry 11

American Mathematical Monthly Elementary Problems, 1932 – 1936

E6.329.S333. (W.R.Ransom) This construction was given in 1525 by Albrecht D¨urer, the great engraver, for a regular pentagon ABCDE, and it is still given in books on mechanical drawing. The circles are all drawn with the same radius, equal to the given length of the side AB, with centers at these points (in order) A, B, Q, C,andE. Calculate the angle ABC to determine whether this is an exact or an approximate construction.

E8.329. (Otto A. Spies) It is required to construct an inscriptible quadrilateral with ruler and compass, given the lengths of the four sides in order.

E11.3210.S334. (W.R.Ransom) Circumscribed about a circle is an isosceles trapezoid, ABCD,inwhichDC < AB,andAD = BC. Two perpendiculars are drawn; DG perpendicular to AB at G,andGH perpendicular to AD at H. Show that DA, DG,andDH are the arithmetic, geometric, and harmonic means, respectively, between the pair of parallel sides AB and DC. YIU : Problems in Elementary Geometry 12

E12.3210.S334. (W.F.Cheney) Two coplanar right triangles, AOC and BOC,havethe common hypotenuse OC. Using vector methods, express the vector OC in terms of the vectors OA and OB.

E15.3210.S337. (Pearl C. Miller) Prove that if two exrternal angle bisectors of a scalene triangle are equal, then the sines of the three interior half - angles form a geometric progression. By external angle bisector is here meant that segment of the line bisecting the exterior angle at a vertex of a triangle, intercepted between that vertex and the opposite side of the triangle.

E16.331.S335. (G.A.Yonosik) Prove that the envelope of the circles whose diameters run from points on a parabola to its focus, is the straight line tangent to the parabola at its vertex.

E17.331. (Wm Fitch Cheney) Of all the right triangles whose areas exceed a million square units and whose three sides are integers without common factor, ﬁnd that one whose perimeter is minimum.

E29.333.S33:493–494. (J.Rosenbaum) The faces of a tetrahedron are congruent triangles whose sides are a, b,andc.If2S = a2 + b2 + c2, show that the volume is 1 (S − a2)(S − b2)(S − c2). 3

E38.335. (J.R.Musselman) It is well known that the midpoints of the sides of any plane quadrilateral constitute the vertices of a parallelogram. Determine the most general condition under which the parallelogram becomes (a) a rhombus, (b) a rectangle, and (c) a square.

E39.335. (Maud Willey) LetCi =0,(i =1, 2, 3) be the equaitons of three circles. Prove 3 that the three circles, i=1 KijCi =0,(j =1, 2, 3), have the same radical center as the three circles Ci =0. Generalization to 3 and higher dimensions.

E40.33(5)296.S34(1)45. (V.F.Ivanoﬀ) A variable circular arc of constant length has one end ﬁxed in position and direction. Find the locus of the other end.

E44.336. (Mannis Charosh) ABC is an isosceles triangle with AB = AC. ADB is a right triangle with D the vertex of the right angle, on the opposite of AB from C.AngleDAB is equal to angle BAC,andDF and CE are perpendicular to AB and AD at F and E respectively. Prove that AF and FB diﬀer by AE. YIU : Problems in Elementary Geometry 13

E45.336. (W.R.Ransom) The ellipse of minimum area which can be circumscribed about a pair of equal, tangent circles, passes through the centers of its largest circles of curvature, and these centers and the two foci are the vertices of a square.

E47.336. (B.H.Brown) By methods of elementary plane geometry construct an equilateral triangle having a vertex upon each of three general lines in a plane, given the position of one vertex. Consider the case when the lines are parallel, and also the case in which the three lines are replaced by three concentric circumferences. What determines the number of solutions in the last case?

E48.337. (Norman Anning) If the squares of the sines of a set of angles are in harmonic progression, show that the squares of the tangents of the same angles are also in harmonic progression.

E52.337. (Moshe Abrahami) Find the area of a triangle in terms of the altitude, interior angle bisector, and median, all from the same vertex of the triangle.

E56.33(8)491.S34(3)189–190. (Otto Dunkel) From the base vertices A and B of an isosceles triangle ABC, segments of straight lines AL and BM of equal length are drawn to the opposite equal sides. Determine by plane geometry the locus of P , the intersection of AL and BM.

E59.338. (J.H.Butchart) In the angle ACB of triangle ABC circles are inscribed tangent respectively to AC at A and to BC at B. Prove that the chords intercepted on the side AB are equal.

E62.339. (W.R.Ransom) Defining a “C−angle” as the figure formed by two internally tangent circles, and its magnitude as the difference of the curvature of those circles, show how to bisect a C−angle geometrically. (If the circles are tangent externally, the magnitude of the C−angle is the sum of their curvatures). If the circles are tagnet to the X−axis at the origin O, and cut the circle x2 + y2 =2x also at P and Q, show that the magnitude of the C−angle equals the difference between the slopes of the chords OP and OQ.

E63.339. (J.Rosenbaum) The bisectors of the interior angles of the triangle ABC meet the sides in the points P , Q, R. Prove that the ratio of the area of the triangle PQR to the area of triangle ABC is 2abc . (a + b)(b + c)(c + a) YIU : Problems in Elementary Geometry 14

E65.33(10).606.S34(5)327–328. (J.M.West) If the vertices of a triangle taken counterclockwise have the abscissas x1, x2,andx3, and if the slopes of the opposite sides are m1, m2, and m3, then prove that the area equals 1 (x − x )(x − x )(m − m ), 2 1 2 1 3 2 3 as well as either of the two similar expressions obtainable from this by cyclic permutation of the subscripts.

E66. Solid geometry

E67.3310. (E.C.Kennedy) Give a scheme for writing down mechanically the sides of an unlimited number of dissimilar right triangles whose sides are integers. After the ﬁrst set, the values are to be written down, not merely indicated, without any calculations whatever. No addition, subtraction, multipication, division, involution or evolution, mental or otherwise, is allowed.

E68.3310. (W.F.Cheney) In the triangle ABC, D is the midpoint of BC. The equilateral triangles ABP , ACQ and ADR are drawn in the plane of triangle ABC,theverticeofeach being listed counterclockwise. Prove that R is the midpoint of PQ. YIU : Problems in Elementary Geometry 15

E70.34(1)44.S34(6)391. (R. MacKay) Show that the area of a right triangle in terms of the bisector of the right angle, t, and the median to the hypotenuse, m, is given by the formulas

2m2t K = √ t2 +8m2 ∓ t where the upper or lower sign is to be used according as t is the bisector of the interior or external angle at the right angle vertex.

E72.34(1)45. (J.M.West) Given that A + B + C = 180circ,provethat sin A cos2 A sin(B − C)=0. cyclic

E.73.34(1)45.S34(6)393–394. (W.F.Cheney) Show that there is just one right triangle whose three sides are relatively prime integers between 2000 and 3000. Answer: (2100, 2059, 2941).

E75.34(2)103. (C.W.Munshower) Show that in any plane triangle the product of the sum of the ratios of the sides to the radii of the corresponding escribed circles, and the ratio of the sum of the sides to the sum of the radii of the escribed circles, is equal to 4.

E86.34(3)189. (R. MacKay) If the faces of a tetrahedron are congruent triangles, prove that the circumcenter and the centroid are coincident.

E93.34(5)327.S34(10)630. (H.T.R. Aude) Find the locus of the centers of the circles in the plane which pass through a given point and are orthogonal to a given circle.

E100.34(6)390. (G.R.Livingston) In two concentric circles, locate parallel chords in the outer circle which are tangent to the inner circle, by the use of compass only, ﬁnd the ends of the chords and their points of tangency.

E102.34(6)390. (R. MacKay) If P is a point on the Euler line of triangle whose sides are a, b, c,onek-th of the distance from the circumcenter O to the orthocenter H,then

9R2 − a2 − b2 − c2 OP 2 = , k2 where R is the circumradius of the triangle. YIU : Problems in Elementary Geometry 16

E107.34(7)447. (J.B.Coleman) A straight line cuts two concentric circles in the points A, B, C, D in that order. AE and BF are parallel chords, one in each circle. CG is perpendicular to BF at G,adnDH is perpendicular to AE at H.ProvethatGF = HE.

E108.34(7)447. (E.Schuyler) Show how to construct a triangle when the orthocenter, the incenter and one vertex are given.

E113.34(8)517. (E.T.Krach) Prove that if three circles are so arranged that their six external tangents are real (each tangent touching two circles), then the three points of intersection of the three pairs of corresponding tangents are collinear.

E121.34(8)577. (W.F.Cheney) The sides of the real triangle ABC are three diﬀerent positive integers, no two of which have a common factor. AD is tangent to the circumscribed circle at A, and meets BC produced at D.PRovethatAD, BD,andCD are each always rational, but that one of them can ever be an integer.

E122.34(9)577. (C.A.Rasmussen) The lines joining the three vertices of a given triangle ABC toapointO in its plane, cut the sides opposite the vertices A, B, C in the points K. L, and M respectively. A line through M parallel to KL cuts BC at V and AK at W .Provethat VM = MW.

E125.34(10)629. (E.Schuyler) Construct the triangle ABC, given the vertex A and the points of contact of BC produced with each of the escribed circles corresponding to sides AC and AB respectively.

E129.34(10)629. (L. Battig) In the parallelogram ABCD points E and F are in sides AB and CD respectively. AF intersects ED in G. EC intersects FB in H. GH produced intersects AD in L and BC in M. Prove by high school geometry that DL = BM.

E220.36?.S372. (C.W.Trigg) If circles be constructed on the sides of a triangle as diameters, show that (a) the common tangent to the circles on two of the sides is the mean proportional between the segments into which the third side is divided by the point of contact of the incircle; and (b) the area of the triangle is equal to the square root of the product of the three common tangents and the semiperimeter. See E296. YIU : Problems in Elementary Geometry 17

E259.3720. (Mannis Charosh) If the tangents of the angles of a plane triangle form an arithmetic progression, prove that the Euler line is parallel to a side of the triangle.

E260.3720. (C.E.Springer) Two lines AB and CD of given lengths slide independently along two ﬁxed skew lines. Show that the locus of the center of the sphere through A, B, C and D is a hyperbolic paraboloid.

E262.3720. (Cezar CO¸snitˇa) Find the locus of the center of a circle which so varies that its radical axes with two ﬁxed circles pass always respectively through two ﬁxed points. YIU : Problems in Elementary Geometry 18

American Mathematical Monthly Elementary Problems, 1940 – 1949

E254.37(1)49. (D.L. MacKay) Given the vertices B and C, and the altitude from A, construct the triangle ABC so that a4 = b4 + c4,wherea, b, c are the sides of the triangle.

E257.37(1)49.S37(8)540. (M. Charosh) Construct the triangle ABC, given the altitude and medain from A, and the diﬀerence b − c of the adjacent sides.

E259.37(2)104.S(). (M. Charosh) If the tangents of the angles are in arithmetic progression, then the Euler line is parallel to one side of the triangle. See also E411, E803.

E262.37(2)104.S37(9)599. (C. Cosnita) Find the locus of the center of a circle which so varies that its radical axes with two ﬁxed circles pass through respectively two ﬁxed points.

E263.37(2)104.S37(9)599–600. (D.L.MacKay) In the triangle ABC, the bisector of angle, the median from vertex B, and the altitude from vertex C are concurrent. Show that the triangle may be constructed with ruler and compasses if the lengths of sides b and c are given.

E265.37(2)104. (W.F. Cheney) A right triangle has integer sides without common factor. When each digit is replaced by a code letter, the sides are SSWTVU, PTWTS and RRWWQ. Solve the code and show that the solution is unique.

E268.37(3)175. (J.E. Trevor) A quadrilateral inscribed in a semicircle consists of three chords and the bounding diameter. Find the radius of the semicircle when the successive chords are of lengths a, b, c. Then particularize when a, b, c are 1, 2, 3 feet respectively.

E269.37(3)175. (C.W. Trigg) If a cevian be drawn to a side of a triangle and circles inscribed in the two triangles thus formed, then (a) the sum of the cevian and the side to which it is drawn is equal to the semiperimeter and the segment between the points of contact of the circles with that side; (b) the product of the radii is equal to the product of the parts into which said segment is divided by the cevain; (c) if the circles are equal, then the area of the original triangle equals the product of the radius by the sum of the cevian and the semiperimeter. YIU : Problems in Elementary Geometry 19

E279.37(5)330.S381. (D.L.MacKay) Given two sides, construct a parallelogram whose angles equal the angles between its diagonals.

E281.37(5)330.S38:51–52. (W.B.Clarke) Let the incircle of triangle ABC touch side a, b, c at points D, E, F respectively. Call the incenter I.WithA as center and AE as radius, swing an arc to cut DI product, inside triangle ABC,atP . Similarly, let arcs centered at B and C cut EI and FI, inside the triangle, at Q and R.LetAP , BQ,andCR meet sides a, b, and c at points J, K,andL respectively. Now prove or disprove the following: (1) AJ, BK,andCL are concurrent; (2) Triangle AJB and AJC, BKA and BKC, CLA and CLB have their incircles equal each to each in pairs.3

E285.37(6)384.S378. (D.L.MacKay) If in triangle ABC,sin2 A +sin2 B +sin2 C =1, prove that the circumcircle cuts the nine-point circle orthogonally.

E293.37(6)479. (J.H.Butchart) Construct three circles through a point P so that the sum 3(1) The lines are not concurrent. (2) The incircles are equal in pairs. YIU : Problems in Elementary Geometry 20 of the directed segments cut oﬀ by the circles on any line through P is zero.

E296.37(7)539. (D.L. MacKay) D, E, F are the centers of the semicircles constructed on the sides BC, CA, AB as diameters, and exterior to the triangle ABC.IFd and e are the lengths of the common external tangents between the points of contact for the semicircles D and E,andD and F , construct triangle ABC,givend, e, and angle A. See E220. (a) d = (s − b)(s − c)etc.

E301.37(9). (D.L.MacKay) If in triangle ABC, B − C =90circ and S, T are the intersections of the internal and external bisectors of angle A with the side BC,prove: b2−c2 (a) sin A = b2+c2 , (b) ST is twice the altitude from A, (c) a2 is the harmonic mean of (b − c)2 and (b + c)2.

E302.37(9). (F.A.Alﬁeri) If A, B, C are the angles of a plane triangle, prove that 1 sin A sin B sin C cot A +cotB +cotC = + + . 2 sin B sin C sin C sin A sin A sin B

E305.37(10)659. (D.L.MacKay) If the external angle bisectors at A and B are equal, must the triangle be isosceles?

E307.37(10)659. (V.Th´ebault) Locate the point P in the plane of the given triangle such that the triangle PAB, PBC,andPCA may have equal perimeters.

E308.37(10)659. (E.H.Clarke) Find the triangle which contains an angle most nearly equal to one radian, from among all possible triangles whose sides are integers of one or two digits.

E311.381. (J.S.Robberson) The quadrant AOB of a circle varies in size and position, but keeps the segment QA of one bounding radius ﬁxed. Find the locus of the point P on the arc AB if it divides that arc in the same ratio as Q divides the radius AO.

E312.381.S389. (D.L.MacKay) If the scalene triangle ABC has its external angle bisectors s−a s−b s−c at B and C equal, show that a is the geometric mean of b and c . See E15.3210.S337.(Pearl C. Miller).

E314.381. (Cezar Co¸snitˇa) Find the locus of the center of a variable sphere which cuts each of two (or three) ﬁxed planes in a circle of constant size. YIU : Problems in Elementary Geometry 21

E361.391.S401. (V.Claudian) The medians of a triangle ABC cut the nine-point circle of that triangle again at D, E,andF , respectively. The tangents to this circle at D, E and F meet the corresponding sides of the orthic triangle (with vertices at the feet of the altitudes of ABC)atthepointsP , Q, R respectively. Prove that P , Q and R are collinear.

E363.391.S407. (D.L.MacKay) Construct triangle ABC,givenA, a,andha + c − b.

E367.392.S402. (C.Co¸snitˇa) The point P moves on the circumcircle of triangle ABC,and the bisectors of angles AP C and AP B meet AC and AB at Q and R respectively. Show that QR passes through I, the center of the circle inscribed in the triangle ABC. Show also that if PS and PT are perpendicular to PQ and PR and cut AC and AB at S and T respectively, then ST passes through the center of the escribed circle which touches side BC between B and C.

E370.392.S402. (V.Th´ebault) Locate the point P within the irregular tetrahedron ABCD so that each of the six planes, each through P and an edge, will bisect the surface of the tetrahedron.

E372.393.S403. (V.Claudian) The variable point Q moves on a circle thorugh the ﬁxed point A,andB is another ﬁxed point in the same plane. The points R and S are the feet of the perpendiculars from A and Q on BQ and AB, respectively. The line through B, parallel to RS, meets AQ at P . Find the locus of P .

E374.393.S403. (D.L.MacKay) What relationshipexists between the sides of a triangle ABC if the bisector of angle A, the median from vertex B, and the altitude from vertex C are concurrent ? Can the three sides be commensurable if the triangle is not equilateral? See also E263.37p600, and E3434.914, apparently without published solution. See R.K.Guy, My favorite elliptic curve: A tale of two types of triangles, Amer. Math. Monthly, 102 (1995) 771 – 781. bc Solution. (C.W.Trigg) By the solution of E263, b cos A = b+c . Hence, 2bc2 2c3 a2 = b2 + c2 − = b2 − c2 + . b + c b + c If the triangle has commensurable sides, and if the proper unit of measurement is chosen, a, b, c will be integers with no common factor. Moreover, b and c must be relatively prime, since any c3 ∗ common factor of b and c would divide a. Hence b+c cannot be an integer, and the only way to make a an integer is to put b = c = 1, in which case the triangle is equilateral. Therefore in all other cases the three sides are incommensurable. ∗ 2c3 This is not correct. b+c can be an integer not divisible by any common divisor of b and c. YIU : Problems in Elementary Geometry 22

122+152−132 5 15 For example, E3434 cites the triangle (13,12,15). Here, cos A = 2·12·15 = 9 = 12+15 . Here, 2c3 b+c = 250 is not divisible by 3.

E379.394.S403. (W.E.Buker) Find a trapezoid whose sides, altitude, diagonals and area are rational.

E380.39p297.S403. (W.F.Cheney) If the radius of a circle is any odd prime p,thereare just two diﬀerent primitive Pythagorean triangles circumscriptible about that circle. Show that, for each such pair of triangles, (a) their shortest sides diﬀer by one; (b) their hypotenuses exceed their corresponding longer legs by one and by two respectively; (c) the sum of their perimeters is six times a perfect square; (d) as p increases without limit, the ratio of their least angles approcaches 2; (e) as p increases without limit, the ratio of their areas approaches 2; (f) the smaller triangle can always be placed inside the larger, so as not to touch it.

E381.395.S406. (W.B.Clarke) Show how to construct a square with one corner on each of four generally placed straight lines in a plane. How many solutions are there in general ? What constitute special cases ? What happens if the lines are placed askew in space ?

E383.395.S404. (Co¸snitˇa) The diameters from the vertices of the triangle ABC,inthe circumscribed circle, cut the opposite sides in E, F and G respectively. L, M and N are the respective midpoints of AE, BF and CG. Show that triangle LMN is homologous to triangle ABC, and that the axis of homology is the orthic axis of the triangle.

E388.397.S405. (V.Th´ebault) On the lateral surface of any right prism, ﬁnd the length of the shortest route from end to end on one lateral edge, winding n times round the prism on the way.

E391.397.S406. (J.Travers) If P is a point inside a square ABCD, so situated that PA : PB : PC =1:2:3,calculatetheangleAP B. Use only the methods of Euclid, Book I. See also E3208, MG1147, MG1418.932.S942.

*** Given three positive numbers a, b,andc, to construct if possible, a square ABCD, together with an interior point P such that

AP : BP : CP = a : b : c.

Solution. Let Q be a point outside the square such that CBQ = ABP and BCQ = BAP.ThenABP ≡CBQ. It follows that and BPQ is a right isosceles triangle, with YIU : Problems in Elementary Geometry 23

√ √ PBQ =90◦,andPQ = 2b. Clearly, then, a, 2b,andc should satisfy the triangle inequality for such a square to exist.

If this√ condition is satisﬁed, we start with a triangle CPQ with CP = c, CQ = a,and PQ = 2b. Outside the triangle, erect a right isosceles PBQ with a right angle at B.The square BCDA onthesideofBC containing P satisﬁes the requirement AP : BP : CP = a : b : c. Let x be the length of BC. By the cosine formula,

x2 + b2 − c2 x2 + b2 − a2 cos PBC = , cos QBC = . 2bx 2bx Since these two angles are complementary,

(x2 + b2 − c2)2 +(x2 + b2 − a2)2 =(2bx)2; ...... [2x2 − (a2 + c2)]2 =4(a2 + c2 − b2)b2 − a4 − c4 +2c2a2. (∗)

If there is nothing wrong here, the right hand side must be nonnegative. Clearly, this quartic√ form cannot be positive deﬁnite. It must have something to do with the condition that a, 2b, and c form a triangle. Indeed,√ it is 162, being the area of triangle CPQ.(Oneeasywayto see this is to replace by 2b by b so that the quartic form becomes

2a2b2 +2b2c2 +2c2a2 − a4 − b4 − c4.

It is easy to that this is zero upon the substitution b = c + a. It follows that c + a − b is a factor of this symmetric polynomial; so are a + b − c and −a + b + c. The remaining linear factor must be a + b + c, and indeed the above polynomial is

(a + b + c)(−a + b + c)(a − b + c)(a + b − c).

Now it is easy to recognize this as 162). Note that 2x2 = AC2 >a2 + c2 since P is an interior point of the square. It follows that the length of a side of the square is 1 x = (a2 + c2 +4). 2 YIU : Problems in Elementary Geometry 24

The distance DP := d can also be determined easily. If, in the above consideration, we replace every occurrence of b by d, we should arrive at the same square. This means that in (*) above,

(a2 + c2 − b2)b2 =(a2 + c2 − d2)d2; ...... (d2 − b2)(d2 − a2 − c2 + b2)=0.

From this, it follows that d = a2 + c2 − b2. HowaboutthecasewhenP is outside the square ?

E394.397.S406. (N.A.Court) If the lines AM, BM, CM joining any point M to the vertices A, B, C of a tetrahedron ABCD meet the repectivel opposite faces in the points P , Q, R, and if the lines DM, DP, DR meet the face ABC in the points S, X, Y , Z,provethat (both in magnitude and in sign) DM 1 DP DQ DR = + + . MS 2 PX QY RZ

E395.397.S406. (Starke) In high school geometry texts and elsewhere one frequently meets the statement that the reason for the straightness of the crease in a folded piece of paper is that the intersection of two planes is a straight line. This is fallacious. What is the correct reason ? Solution. (L.R.Chase) Let P , P be two points of the paper that are brought into coincidence by the process of folding. Then any point A of the crease is equidistant from P , P ,since the lines AP , AP are pressed into coincidence. Hence, the crease, being the locus of such points A, is the perpendicular bisector of PP.

E396.398.S407. (D.L.MacKay) Given a triangle ABC, construct a point X such that the three lines drawn through X, each parallel to a side of the triangle and limited by the other two sides, are equal.

E398.398.S408. (V.Claudian) Given a triangle ABC,letO be the circumcenter, A the projection of A on BC, M any other point of BC,andB1, C1 the respective projections of B, C on AM. Let lines through M,paralleltoA C1 and A B1, meet AC and AB in points P and Q respectively. Prove that the lines PQ and OM are perpendicular.

E400.398.S407. (H.S.M.Coxeter) Show how to dissect a regular hexagon by straight cuts into the smallest possible number of pieces which can be reassembled to form an equilateral triangle (of the same area). YIU : Problems in Elementary Geometry 25

E405.401.S409 (J.Travers) Construct points P and Q on the respective sides AB and BC of a given triangle ABC,sothatAP = PQ = QC.

E407.401(correction 406).S409. (V.Claudian) Let A, B C be the feet of the altitudes of a triangle ABC,andH the orthocenter. Let the parallels through H to BC, CA, AB meet BC, CA, AB in DD, E, F respectively; and let the parrllels through H to BC, CA, AB meet BC, CA, AB in D, E, F . Prove that the six points D, E, F and D, E F lie respectively on two parallel lines, perpendicular to the Euler line.

E409.401.C406. (V.Th´ebault; withdrawn) Consider a hexagon whose vertices are the ends of three diameters of a circle. Show that the sum of the products of the distances of a variable point on the circle from pairs of opposite sides of the hexagon is constant. (False).

E410.402.S.() What are the smallest positive integers a, b, c which are the sides of a triangle whose medians are also integers? (Partial solution by W.E.Bueker) The problem of ﬁnding triangles whose sides and medians are integers is an old one. (See, Dickson’s History, vol.2, pp.202–205). Particular solutions were obtained by Euler and rediscovered many times, the simplest being 174,170,136 for the sides and 127, 131, 158 for the medians. (A recent account is to be found in Alliston, Mathematical Snack Bar, pp.24–25.) While these investigations do not seem to prove that the above solution is the smallest one, I suggest that otherwise the problem is scarely elementary. Editor’s note: The squares of the medians are easily seen to be

−a2 +2b2 +2c2, −b2 +2c2 +2a2, −c2 +2a2 +2b2,

a b c where a = 2 , b = 2 , c = 2 .Euler(Operr postuma, vol.1), 1862, pp.102–103) observed that the values

a =(m + n)p − (m − n)q, b =(m − n)p +(m + n)q, c =2|mp − nq| make the third median 2(np + mq), and that the other medians are integers too if p =(m2 + n2)(9m2 − n2), and q =2mn(9m2 + n2), m, n being integers subject to certain inequalities. Discarding any common factor of p and q, we obtain the following simple cases. (But other expressions for p and q might provide hitherto unknown solutions). YIU : Problems in Elementary Geometry 26

mn pq Halfsides Medians 1 2 25 52 127, 131, 158 261, 255, 204 2 1 175 148 377, 619, 404 975, 477, 942 3 1 200 123 277, 446, 477 881, 640, 569 2 3 13 20 85, 87, 68 131, 127, 158 5 3 68 65 207, 328, 145 463, 142, 529

E411.403.S4010. (J.H.Butchart) Prove that, if the sides of a triangle form an arithmetic progression, the line joining the centroid to the incenter is parallel to one side. See also E803 and E259.

E415.403.S4010. (C.Co¸snitˇa) In a triangle of sides a, b, c prove that the distance from the centroid to the incenter I is given by the formula 3(a + b + c)GI = a2(b − c)2 − (b2 + c2 − a2)(c − a)(a − b).

E417.404.S411.(J.F.Kenney) Let E be any point outside a circle, ABE the diameter through E,andCDE any chord through E. In the triangle BCE, show that the side CE is divided by D into segments CD and DE whose ratio is less than the ratio of the angles E and C.

E418.404.S411.(W.E.Buker) Find triangles with rational sides and angle bisectors. Solution. (E.P.Starke) This problem has already been solved as part of E331. Following the notation and analysis given there, the internal angle bisectors have lengths bc A ca B ab C 2 cos , 2 cos , 2 cos . b + c 2 c + a 2 a + b 2 The sides being rational, it is necessary and suﬃcient that the three half-angles have rational cosines. It is then easy to establish that they must have rational sines. [Denote the incenter by I. We may assume IA, IB, IC rational. Now, A + B AIB = π − ( ). 2 The area of triangle AIB being 1 1 C IA · IB · sin AIB = IA · IB · cos , 2 2 2 is rational. Similarly, the areas of triangles IBC and ICA are rational. It follows that the sines A B C of 2 , 2 , 2 are all rational.] YIU : Problems in Elementary Geometry 27

It is then easy to establish that they must have rational sines. [Denote the incenter by I. We may assume IA, IB, IC rational. Now, A + B AIB = π − (( ). 2 The area of triangle AIB being 1 1 C ( IA · IB · sin AIB =( IA · IB · cos( , 2 2 2 is rational. Similarly, the areas of triangles IBC and ICA are rational. It follows that the sines A B C of 2 , 2 , 2 are all rational.]

Theorem The internal angle bisectors of a triangle with rational sides are all rational if and A B C only if tan 4 ,tan4 ,andtan4 are all rational.

Theorem If the sides and internal angle bisectors are rational, so also are the external angle bisectors, the altitudes, the area, and the ﬁve radii.

A Let u1 := tan 4 etc. It is easy to see that

1 − u1u2 − u2u3 − u3u1 = u1 + u2 + u3 − u1u2u3, from which 1 − u1 − u2 − u1u2 u3 = . 1+u1 + u2 − u1u2 Two positive rational numbers u1, u2 < 1 determine u3 < 1 if and only if u =(1+u1)(1+u2) < 2. In this case, 2 − u 0

Triangles with rational angle bisectors

(u1,u2,u3) (t1,t2,t3) (a, b, c) (wa,wb,wc) 1 1 1 4 8 13 27300 4641 20400 ( 2 , 4 , 13 ) 3 , 15 , 84 ) (289, 250, 91) ( 341 , 38 , 77 ) 1 1 2 3 8 36 400 11781 1700 ( 3 , 4 , 9 ) ( 4 , 15 , 77 ) (289, 250, 231) ( 481 , 52 , 7 ) 1 1 1 4 5 16 12600 26208 975 ( 2 , 5 , 8 ) ( 3 , 12 , 63 ) (169, 125, 84) ( 209 , 253 , 7 ) 1 1 3 3 5 33 30800 48048 2600 ( 3 , 5 , 11 ) ( 4 , 12 , 56 ) (169, 125, 154) ( 279 , 323 , 21 ) 1 1 5 8 5 140 101745 248976 2652 ( 4 , 5 , 14 ) ( 15 , 12 , 171 ) (338, 289, 399) ( 344 , 737 , 11 ) YIU : Problems in Elementary Geometry 28

See also AMM 46 (1939) pp.345 – 347.

E1384.59?.S604. (J.H.Butchart) Construct a circle through two given points, separated by a given circle, which shall cut the given circle at the smallest possible angle.

E1394.59?.S606. (V.Th´ebault) Given a trihedral angle and a point within it, construct the plane through the point which intecepts on the trihedral angle the tetrahedron of minimum volume. E1397.601.S607. (Bankoﬀ) Show that AI ≤ AH.

E1398.601.S607. (A.N.Aheart) If A, B, C are the angles of a triangle, show that

cos A +cosB +cosC<2.

≤ 3 Indeed, cos A +cosB +cosC 2 , with equality for equilateral triangles. P.D.Thomas quoted Euler’s relation d2 = R(R − 2r)and r cos A +cosB +cosC =1+ R from Johnson (p.191). David Zeitlin applied the Erd¨os - Mordell inequality to the circumcenter. Also,

3 d2 cos A = − . 2 2R2

E1402.602.S608. (F.Leuenberger) Let r denote the radius of the inscribed sphere of a tetrahedron T and let ri, I =1, 2, 3, 4, denote the radii of the exspheres of T which touch one 4 ≥ face of T and the other three faces of T produced. Show that i=1 ri 8r, with equality if and only if T is isosceles.

E1406.603.S609. (M.Goldberg) Cut an obtuse triangle into the least number of acute triangles.

E1417.605. (R.Hartop) Given a unit circle with point P on the circumference and a distance d1,0

E1411.604.S6010. (Shee) Let ABCDE be any pentagon inscribed in a circle and let P , Q, R, S, T be intersections of the diagonals such that P and Q lie on AC, Q, R on BD, R, S on CE,andS, T on DA.Provethat AB · BC · CD · DE · EA AP · BQ · CR · DS · ET = . AC · BD · CE · DA · EB CP · DQ · ER · AS · BT

E1420.605. (Th´ebault) Let A, B, C (A,B,C) be the centers of squares described exteriorly (interiorly) on the sides BC, CA, AB of a triangle ABC. Show that the radical center of the circles A(A), B(B), C(C)(A(A),B(B),C(C)) coincides with the nine-point center of triangle ABC.

E1425.605. (D.J.Newman) If a square lies within a triangle, prove that the area of the square does not exceed half the area of the triangle.

E1427.607. (F.Leuenberger) In a triangle, √ 3(a + b + c) ≥ 2(ha + hb + hc), with equality if and only if the triangle is equilateral.

E1433.608. (A.Oppenheim) Let P be apoint in the interior of a triangle and let the distances from the vertices of the triangle be x, y, z, and from the sides of the triangle be p, q, r. Show that xyz ≥ (q + r)(r + p)(p + q).

E1436.609. (M.K.Shen) Through the vertices of a given triangle ABC draw straight lines , m, n respectively, such that n and intersect in D, and m in E, m and n in F inside the triangle and 1 (a) ABE = BCF = CAD = DEF = 4 ABC. (b) DEF is similar to ABC and has an area equal to a given fraction of ABC. YIU : Problems in Elementary Geometry 103

American Mathematical Monthly, Advanced Problems, 1950 – 1959

AMM4456.51?.S531.(V.Th´ebault) If through the vertices A, B, C, D of a tetrahedron parallel planes are drawn cutting a given line L in points A2, B2, C2, D2 and if A1, B1, C1, D1 are the points in which the lines AA2, BB − 2, CC2, DD2 cut the planes BCD, CDA, DAB, ABC,then AA BB CC DD 2 + 2 + 2 + 2 =2. AA1 BB1 CC1 DD1

AMM4465.51?.S532.(E.I.Gale) AXBZ is a jointed rhombus with sides of length 4a.(See ﬁgure. For convenience, a is taken considerably greaer than half the unit). AO and BO are bars 1 of equal length. The ﬁxed centers are O and O with OO half a unit. Let O D = 2 , FE =4a, FZ = HG = BE =2a, HZ = FG = a. D is an adjustable set screw on bar FZD so that the length ZD can be set at pleasure. 1 Show that as D moves in a circle about O , X describes the general conic of eccentricity ZD.

AMM4470.52?.S534.(V.Th´ebault) In a triangle ABC, three lines a, b c drawn through the vertices A, B, C determine by their intersections a triangle ABC and their isogonals a, b, c determine a triangle ABC. (1) Show that the orthic triangles of ABC and ABC have equal perimeters. (2) If, further, a, b, c are equally inclined to AB, BC, CA, show that the circles ABC and ABC are symmetric with respect to the line joining the symmedian point to the circumcenter.

AMM4477.52?.S535.(C.E.Springer) Show that the area of the Morley triangle of a triangle is ( sin2 A )(1 − 4 cos2 A +16 cos A ) 3 3 3 · π−A sin A sin 3 times that of the given triangle. See also AMM1943 p.552.

AMM4485.52?.S536.(S.T.Kao) Through the orthocenter H of a triangle ABC draw any pair of perpendicular lines l1 and l2 and let A1,a2; B1,B2; C1,C2 be the respective points of intersection with the three sides BC, CA, AB. Show that the three poiints P , Q, R which divide the three segments A1A2, B1B2, C1C2 in the same ratio r lie on a line. YIU : Problems in Elementary Geometry 104

AMM4491.52?.S538.(C.S.Venkataraman) If ω denotes the Brocard angle of a triangle ABC,provethat √ (1) the sides are equal when cot ω = 3; (2) the squares of the lengths of the sides are in arithmetic progression when cot ω =3cotB.

AMM4500.52?.S539.(J.R.Musselman) Given three points Ai, i =1, 2, 3, and a line L cut by the lines AjAk making angle αi in thepositive sense. Show that the lines drawn through Ai making angles π − αi, in the positive sense, with L are concurrent at a point P on the circumcircle of A1A2A3. Further, if the altitudes with Ai make angles βi with L then the lines through Ai making angles π − βi with L are concurrent on theecircumcircle of A1A2A3 at a point Q diametrically opposite to P .

AMM4530.533.(V.Th´ebault) In a tetrahedron ABCD,letA, B, C, D be the feet of the altitudes AA, BB, CC, DD. The planes drawn through the midpoints of BC, CA, AB, DA, DB, DC perpendicular to BC, CA, AB, DA, DB, DC respectively, are concurrent at apoitP , which is the radical center of the spheres described with the vertices A, B, C, D as centers and with the altitudes AA, BB, CC, DD as radii.

AMM4540.535.(J.Gallego-Diaz) Determine the equaltion of the most general caurve such that the locus of the centers of equilateral triangles inscribed in it is the same curve. (The equilateral hyperbola is a particular case).

AMM4549.537.S549. (R.Obl´ath) The Gauss-Newton line of the complete quadrilateral formed by the four Feuerbach tangents of a triangle is the Euler line of the triangle.

AMM4558.539.(V.F.Ivanoﬀ) If an octagon l1l2 ···l8 is inscribed in a conic, then the eight points of intersection of the sides li and lj, j ≡ i + 3 mod 8, lie on another conic.

AMM4562.539.S552. (Th´ebault) In a triangle ABC,letP be a point having normal − x coordinates (x, y, z) and consider the points A , B , C with normal coordinates ( 2 ,y,z), − y − z (x, 2 ,z)and(x, y, 2 ). (1) The points A, B, C, A, B, C lie on one conic S, and there is a onic with respect ot which the triangles ABC and ABC are self polar. (2) If AP , BP, CP cut BC, CA, AB in A1, B1, C1 and if A P , B P , C P cut B C , C A , A B in A1, B1, C1, the triangles ABC and A B C are circumscribed about a conic Σ, the points of tangency being A1,B1,C1 and A1, B1, C1. (3) The conics S and Σ have double contact along the common polar of P with respect to these conics. YIU : Problems in Elementary Geometry 105

AMM4600.547.S5510. (Bankoﬀ) Vertices A − C and B − D of squares ABCD are joined by quadrants of circles (B)and(C). A semi-circle (O1) is described internally on the diameter BC and a circle (O2) is drawn tangent to the three arcs. Another circle (O3)isdrawntangent to circle (O2)andtoarcsAC and BC, and a right triangle is formed joining O3 and O1 and dropping a perpendicular from O3 upon BC. Successively tangent circles are drawn in the same manner (with (On) tangent to (On−1)andtoarcsAC and BC, and right triangles are formed (with O1On for hypotenuse). SShow that the inﬁnitude of triangles so construced are Pythagorean.

AMM4611.549,551(corrected). (Thébault) Given five spheres, if one of them is orthogonal to the four others, then the centers of the four are the vertices of an orthocentric tetrahedron whose orthocenter coincides with the center of the fifth sphere.

AMM4630.553.S566. (Th´ebault) Having given two confocal conics C1 and C2,letM1 on C1 and M2 on C2 be so chosen that the tangents at these points are perpendicular. Show that the envelope of M1M2 is another conic having the same foci and having asymptotes passing through the intersections of C1 and C2.

AMM4651.557.S569. (J.Rosenbaum) On the sides of a parallelogram A1A2A3A4 ,equilateral triangles AiAi+1Bi are constructed exteriorly. Then equilateral triangles BiBi+1Ci are constructed interiorly to B1B2B3B4.ProvethatC1C2C3C4 coincides with A1A2A3A4. Generalize to the case when the original parallelogram is replaced by a polygon of n sides.

AMM4669.561.S572. (M.Goldberg) (1) What is the relation connecting the lengths of the sides of rectangular skew (spatial) pentagon? (2) Among all possible rectangular skew pentagons, what are the lengths of the sides of the pentagon in which the ratio of the longest to the shortest is a minimum?

AMM4679.563.S573. (H.Demir) If A1A2A3A4A5 is a cyclic pentagon and if Ωij denotes theorthopoleofthelineAiAj with respect to the triangle formed by the remaining three vertices, then prove that the ten points Ωij all lie on a circle.

AMM4690.564. (Th´ebault) Being given a tetrahedron ABCD and the tetrahedron A1B1C1D1 obtained by passing planes through A, B, C, D parallelt to the opposite faces of ABCD,show that 2 2 2 − 2 − 2 PA + PB + PC 2PD PD1 is a constant independent of the position of point P . Extend this property to a skew polygon of n vertices. YIU : Problems in Elementary Geometry 106

AMM4700.567. (J.W.Clawson) It is well known that the midpoints of the diagonals of a 4-lone lie on a straight line which has been called the Newtonian of the 4-line. It is also well known that the Newtonians of the ﬁve 4-lines obtained by omitting in turn each of the sides of a 5-line concur in a point which we may call the Newtonian point of the 5-line. Prove

1. that the Newtonian points of the six 5-lines obtained by omitting in turn each of the sides of a 6-line lie on a conic which may be called the Newtonian conic of the 6-line;

2. that the Newtonian conics of the seven 6-lines obtained by omitting in turn each of the sides of a 7-line concur in three points, two of which may be imaginary.

AMM4710.569. (H.Demir) Prove that if in a complete quadrangle inscribed in a circle (O) one pair of opposite sides are isotomic lines with respect to a triangle inscribed in (O), then the remaining pairs of opposite sides are also isotomic lines with respect to the same triangle.

AMM4718.571. (V.F.Ivanoﬀ) The six points of intersection of a conic and a cubic determine a Pascal haxagon. Show that the residual six points of intersection of the sides of the hexagon with the cubic form two collinear sets, and the lines determined by these sets meet on the Pacal line.

AMM4726.572. (Th´ebault) If the parallels to the asymptotes of a conic (C), drawn through an arbitrary point P of its plane, intersect (C)inP1 and P2, if the perpendiculars to PP1 and PP2 at P1 and P2 intersect in a point O, and if the polar of P with respect to (C) intersects the conic in M1 and M2, then the perpendicular bisector of segment M1M2 passes through O.

AMM4730.573. (E.J.F.Primrose) If a ﬁnite set of points in complex 3-dimensional space has the property that the line joining any two points of the set passes through a third point of th set, must all points of the set be coplanar?

AMM4908.605. (J.Rainwater) Consider a triangle abc divided into four smaller triangles, a central one def inscribed in abc and three otherson the three sides of def. Show that def cannot have the smallest area of the four unless all four are equal with d, e, f the midpoints of the sides of abc. YIU : Problems in Elementary Geometry 107

American Mathematical Monthly Elementary Problems, 1976 – 1991.

AMM.741,p61.() A generalization of Morley’s theorem. Coxeter-Greitzer p.49,163: Morley’s theorem as a consequence of the fact that A1A2, B1B2 and C1C2 are concurrent

E1073.53,417.() A polygonal spiral A1A2A3 ··· of unit segments winds counterclockwise and is construcrted in the following manner: Point A1 is at the origin, point A2 is at (1, 0), 2π ≥ An−1AnAn+1 = n for all n 2. Is there a point lying within the interior of each An−1AnAn+1? If so, what are its coordinates?

E1822.65?.S777.(N.Ucoluk) Let A, A1 and B,B1 be any two pairs of points in the plane. Consider the locus of points N such taht the angles ANA1 and BNB1 (with measures having absolute values α and β respectively) satisfy the codition α = kβ,wherek is a given positive real number. (a) Determine the diﬀerentiability properties of this locus, and (b) when the tangent line exists give a geometric procedure (ﬁnite) for its construction.

E2319.72?.S761.(C.S.Ogilvy) Find the side of the largest cube that can be wholly contained with a tetrahedron of side.

E2358.755.() Let ABC be a triangle. If X is a point on side BC,letAX meet the circumcircle of ABC again at X. Prove or disprove: if XX has maximum length, then AX lies between the median and the internal angle bisector issuing from A.

E2401.732.S763.(V.F.Ivanoﬀ) The exterior angle bisectors of a convex polygon P0 form a polygon P1, whose exterior angle bisectors form a polygon P2, and so on. Prove that Pn approaches a regular polygon as n →∞. YIU : Problems in Elementary Geometry 108

E2407.73?.S74? (A.W.Walker) Given the circumcenter O, the orthocenter H,andthe incenter I of an unknown triangle (T ), (a) locate by euclidean construction the Gergonne point and the Lemoine point of (T ), (b) locate the orthocenters of the pedal triangles of H and I. Editorial Note: This problem is interesting because triangle (T ) cannot in general be constructed from the given points, but many points related to (T ), including those mentioned in this problem, can be so constructed. The two solutions received are quite involved, so we do not take the space here to print either of them.

E2453.741.S752.() Determine all rational numbers r for which 1, cos 2πr and sin 2πr are linearly dependent over the rationals.

E2462.743.S855.(H.Demir) Let P be an interior point of ABC. Erd¨os -Mordell inequality: R1 + R2 + R3 ≥ 2(r1 + r2 + r3). Prove that the above inequality holds for every point P in the plane of ABC when we make the interpretation Ri ≥ 0alwaysandri is positive or negative depending on whether P and A are one the same side of BC or on opposite sides. Solution by Dodge appeared in CM10.p274-281.

E2471.74.S755.() Let ma, wa and ha denote the median, angle bisector and altitude to side a of ABC respectively. Show athat

(b + c)2 m b2 + c2 m ≤ a , ≤ a . 4bc wa 2bc ha When does equality hold?

E2475.74.S756.() Under what conditions can the four tritangent circles of a triangle be rearranged so as to be mutually tangent?

E2477.74.S756.() A straight line L meets the sides BC,CA and AB of ABC with orthocenter H at X, Y , Z respectively. DE is a diameter of the circle ABC Through X, Y , Z lines BC, CA and AB are drawn parallel to AE, BE CE to form a triangle ABC oppositively similar to ABC.IfD, E H are the images of D, E, H for this similarity, prove that in general (a) The lines AA, BB CC, DD, HH concur, so that ABCD and ABCD (ABCH and ABCH) are oppositely similar perspective cyclic (orthocentric) quadrangles; (b) The lines DH and HD meet at the invariant point of the similarlity, and DHDH is a cyclic quadrangle; (c) The axis L is perpendicular to DD and bisects EE. YIU : Problems in Elementary Geometry 109

E2489.74?.S765.() Given ABC, ﬁnd the locus of all points P not necessarily in the plane of ABC, with the property that the three triangles PAB,PBC,PAB have the same area.

E2498.74?.S765.(R.E.Smith) Given triangle ABC, ﬁnd the locus of all points R (not necessarily in the plane of ABC) with the property that the three angles RAB, RBC and RCA have the same area.

E2501.74.S759.() Let ABC be a triangle with C ≥ B ≥ A. Show that BIO is a right triangle if and only if a : b : c =3:4:5.

E2503.74?.S761.(R.F.Jackson) AﬁxeddiskC0 of unit radius is centered at (−1, 1). Be- ginning with the disk C1, centered at (1, 1) and tangent to the x−Axis and to C0, an inﬁnite chain of disks {Ck} is constructed, each tangent to the x−axis, to C0,andtoCk−1.Findthe sum of their areas.

E2504.74?.S761.(Garfunkel) Prove or disprove √ 3 h + m + t ≤ (a + b + c). a b c 2 See also MG752. Editor’s remark: Lu Ting and Richard Lo obtain the following generalization: √ 1 t + m + t 3 ≤ a b c ≤ ; 2 a + b + c √2 1 h + m + t 3 ≤ a b c ≤ ; 4 a + b + c 2 3 t + m + m ≤ a b c ≤ 1. 8 a + b + c

E2505.74?.S761.(Garfunkel) Extend the medians of a triangle to meet the circumcircles again, and let these chords be Ma, Mb, Mc respectively. SHow that See also E2959.

4 M + M + M ≥ (m + m + m ); a b c 3 a b c 2√ M + M + M ≥ 3(a + b + c). a b c 3 When does equality occur ? YIU : Problems in Elementary Geometry 110

E2512.751.S763.() Let T1 and T2 be two triangles with circumcircles C1 and C2 respectively. Show that if T1 meets T2 then some vertex of T1 lies in (or on) C2 or vice versa. Generalize.

E2513.751.S762.(N.Felsinger) Let P be a simple (non-self-intersecting) planar polygon. If A is a point in th plane, and if E is an edge of P ,thenE is viewable from A if for every point x of E, the line segment joining A to x contains no point of P other than x. (a) Let A and P be arbitrary. Must some edge of P be viewable from A ?Examinethe cases of A exterior to P and interior to P separately. (b) Find suﬃcient conditions of A in order that some edge of P is viewable from A.

E2514.751.S763.(G.A.Tsintsifas) Let P be a convex polygon and let K be the polygon whose vertices are the midpoints of the sides of P . A polygon M is formed by dividing the sides of P (cyclically directed) in a ﬁxed ratio p : q where p + q = 1. Show that

|M| =(p − q)2|P | +4pq|K|, where |M| denotes the area of M etc.

E2517.752.S763.(A.G.Ferrer) Let P denote a point interior to the triangle ABC,andlet r1,r2,r3 denote the distance from P to the sides of the triangle. If p denotes the perimeter of the pedal triangle, show that C (r + r )cos ≤ p. 1 2 2 When does equality occur ?

E2531.754.S766.(V.F.Ivanoﬀ) Given points A, B, C, D, E, F in the plane, let ABC denote the directed area of triangle ABC,provethat

AEF · DBC + BEF · DCA + CEF · DAB = DEF · ABC.

E2542.741.S752,767.(J.Anderson) Starting with an arbitrary convex polygon P1,ase- quence of polygons is generated by successively “chopping off corners”; thus if Pi is a k−gon, − then Pi+1 is a (k +1) gon, etc. At the jth step, let dj be the altitude of the cut-off triangle, measured from the cut-off vertex. Prove or disprove: The series dj converges.

E2553.758.S771.(V.B.Sarma) Suppose that A, B, C, D are concyclic and that the Simson line of A with respect to triangle BCD is perpendicular to the Euler line of triangle BCD. Show that the Simsion line of B will be perpendicualr to the Euler line of triangle CDA.Istheabove result true if we replace ‘perpendicular’ by ‘parallel’ ? YIU : Problems in Elementary Geometry 111

E2557.758.() Find all cyclic quadrilaterals with integral sides, each of which has its perimeter numerically equal to its area. The following refereences may be of interest E1168[1955,365;1956,43], E2420[1973,691;1974,662]; M.V.Subbarar, Perfect triangles, AMM78(1971),384-385; Marsden, Triangles with integer- val- ued sides, AMM81(1974),373-376.

E2566.75.S773.(E.Kramer) A triple of natural numbers is called an obtuse Pythagorean triple if they are the sides of a triangle with an angle 120◦. Such a triple is primitive if they have no common factor other than 1. (i) Show that each positive integer except 1, 2, 4, 8 can appear as the smallest member of an obtuse Pythagorean triple. (ii)* What positive integers can appear in primitive obtuse Pythagorean triples? Answer. (ii): either an odd number > 3 or a multiple of 8. There is an analogous notion of acute Pythagorean triples (requiring the triangle to be nonequilateral). If (a, b, c)isan OPT, then (a, c, a + b) is an APT, and all APT can be obtained in this way. Using this it is easy to show that (i) holds also for APT’s.

E2576.752.S775.(R.L.Helmbold) What is the area of the orthogonal projection of the x 2 y 2 z 2 ellipsoid ( a ) +(b ) +(c ) = 1 onto a plane perpendicular to the unit vector Gn =(n1,n2,n3)?

π E2579.762.S776.(B.Klein and B.White) Let 0 <θ< 2 and let p, q be arbitrary distinct points in the euclidean plane E. Deﬁne fθ(p, q) to be the unique point r in E such that triangle pqr is in the counterclockwise sense and rpq = rqp = θ radians. Show that fπ/3(p, q)can be written as an expression involving only fπ/6, p, q, and parentheses.

E2584.762.S776.(Coxeter) Describe an inﬁnite complex congruent isosceles triangles, ex- tending systematically throughout three-dimensional euclidean space in such a way that each side of every triangle belongs to just two other triangles.

E2585.762.S776.(J.Mycielski) Prove that for every triangulation of a 2-dimensional closed surface, the average number of edges meeting at a vertex approaches 6 in the limit as the number of triangles used approaches inﬁnity.

E2617.769.S781.(E.Ehrhart) A convex body is cut by three parallel planes. If the three sections thus produced have the same area, show that the portion of the body lying between the two outside plane is a cylinder. Does the same conclusion follow if instead we are given that the three sections have the same perimeter ?

E2625.7610.S782.(H.Demir) Let Ai, i =0, 1, 2, 3(mod4),befourpointsonacircleΓ. YIU : Problems in Elementary Geometry 112

Let ti be the tangent to Γ at Ai and let pi and qi be the lines parallel to ti pasing through the points Ai−1 and Ai+1 respectively. If Bi = ti ∩ ti+1,andCi = pi ∩ qi+1, show that the four lines BiCi have a common point.

E2630.771.S784.(E.T.Ordman) Suppose that a polyhedral model (made, say, of card- board) is slit along certain edges and unfolded to lie flat in theplane. The cuts may not be made so as to disconnect the figure. Now suppose that the resulting plane figure is again folded up to make a polyhedron (folding is allowed only on the original lines). The new polyhedron is not necessarily congruent to the original one. Find some interesting examples.

E2632.771.S784.(A.Rosenfeld) Define the discrepancy d(A, B) between two plane geometric figures to be the area of their symmetric difference. Let A be a circle of radius r. Determine the inradius of the regular n−gon B for which d(A, B) is minimal.

E2634.771.S784.(Garfunkel) Let Ai, i ≡ 0, 1, 2 (mod 3), be the vertices of a triangle, Γ its inscribed icrcle with center I.LetBi be the intersection of the segment AiI of the segment with Γ and let Ci be the intersection of the line AiI with the side Ai−1Ai.Provethat AiCi ≤ 3 AiBi.

E2639.772.S785.(G.A.Tsintsifas) Let ABC be a triangle with A =40◦, B =60◦.Let D and E be points lying on the sides AC and AB respectivley, such that CBD =40◦ and BCE =70◦.LetF be the point where the lines BD and CE intersect. Show that the line AF is perpendicualr to the line BC.

E2641.773.S786.(P.Straﬃn) Given a convex polygon, and a point p inside it, deﬁne D(p)to be the sum of perpendicular distance from p to the sides of the polygon (extended if necessary). Characgterize those convex polygons for which D(p) is independent of p.

E2646.772.S786.(W.Wernick) Let A1,...,An be vertices of a regular n−gon inscribed in a circle with center O.Let B be a point on arc A1An and θ = AnOB.IFak is the length of n − k the chord BAk,express k=1( 1) ak as a function of θ.

E2649.774.S787.(A,Oppenheim) Let ABC be a non-obtuse triangle, with angles measured in radians. Show that ≤ a b c (1) 3(a + b + c) π( A + B + C ); 2 2 2 ≥ a2 b2 c2 (2) 3(a + b + c ) π( A2 + B2 + C2 .

E2657.775.S788.(G.Tsintsifas) Let A = A0A1 ···An and B = B0B1 ···Bn be regular YIU : Problems in Elementary Geometry 113 n−simplices in Rn. Assume that the ith vertex of B lies on the ith face of A,0≤ i ≤ n.What is the minimal value of their similarity ratio ?

E2660.776.S788.(E.Ehrhart) A quadrilateral is cyclic if its vertices lie on a circle. Find the number of congruence classes of cyclic quadrilaterals having integer sides and given perimeter n. See also AMM796.p477.

E2668.777.S7810.(R.Evans and I.M.Issacs) Find all non-isosceles triangles with two or more rational sides and with all angles rational (measured in degrees). Solution. Such a triangle must have angles 30◦, 60◦ and 90◦.

E2669.777.S7810.(I.J.Schoenberg) Let a>b>0. For a given r,0

··· ··· − E2674.778.(G.Tsintsifas) Let S = A0A1 An and S = A0A1 An be regular n simplices such that Ai lies on the opposite face of Ai. Is it true that the centroids of S and S coincide ?

E2680.779.S792.(J.W.Grossman) Let ABCD be a convex quadrilateral in the hyperbolic plane. Assume that AD = BC and that

A + B = C + D.

Does AB = CD follow from the above hypotheses ? (It does in the euclidean plane).

E2682.779.S793.(D.Hensley) Let E be an ellipse in the√ plane whose interior area A ≥ 1. Prove that the number n of integer points of E satisﬁes n<6 3 A.

E2687.7710.S799.(R.Evans) Does there exist a triangle with rational sides whose base equals its altitude ? Answer. No.

E2694.781.S796.(I.J.Schoenberg) Let Π be a prism inscribed in a sphere S of unit radius and center O. The base of Π is a regular n−gon of radius r. For each face F of Π dropa directed ∗ perpendicular from O and let AF be the point where it intersects S.LetΠ be the polyhedron obtained by adding to Π, for each face F , the pyramid of base F and apex AF . For which values of r is Π∗ convex ?

E2701.783.S795.(R.Stanley) Find the volume of the convex polytope determined by xi ≥ 0, YIU : Problems in Elementary Geometry 114

1 ≤ i ≤ n and xi + xi+1 ≤ 1, 1 ≤ i ≤ n − 1.

E2715.785.S798,804.(Garfunkel) Let G be the centroid of ABC. Prove or disprove 3 sin GAB +sinGBC +sinGCA ≤ . 2 The inequality is true.

E2716.785.S828.(Garfunkel) Let P be an interior point of triangle ABC.LetA,B,C be the points where the perpendiculars drawn from P meet the sides of ABC.LetA,B,C be the points where the lines joining P to A, B, C meet the corresponding sides of ABC.Prove or disprove that AB + BC + CA ≤ AB + BC + CA.

3 E2727.787.S799.(D.P.Robbins) Two triangles A1A2A3 and B1B2B3 in R are equivalent if there exist three diﬀerent parallel lines p1,p2,p3 and rigid motions σ, τ such that σ(Ai)and τ(Bi) lie on pi, i =1, 2, 3. Find necessary and suﬃcient conditions for equivalence of two triangles.

E2728.787.S799.(J.G.Mauldon) Let A, b, c, d be radii of four mutually externally tangent right circular cylinders whose axes are parallel to the four principal diagonals of a cube. Char- acterize all quadruples a, b, c, d which arise in this way.

E2736.788.S822.(E.Ehrhart) Let be a closed triangle and P0,A0,P1,A1,... an inﬁnite sequence of points in a plane. Assume that Pi = Pi+1, Ai = Ai+1,eachAi is a vertex of and the midpoint of the segment [Pi,Pi+1], and that [Pi,Pi+1] ∩= {Ai}.ProvethatPn = P0 for some positive n.

E2740.789.S858.(V.Pambuccian) Show that if P is a convex polyhedron, one can ﬁnd a square all of whose vertices are on some three faces of P , as well as a square whose vertices are on four diﬀerent faces of P .

E2746.7810.S801.(G.F.Shum) Let A1,A2,...,An be distinct non-collinear points in the plane. A circle with center P and radius r is called minimal if AkP ≤ r for all k and equality holds for at least three values of k. If A1,...,An vary, n being ﬁxed, what is the maximum number of minimal circles ?

In Poincar´e’s half-plane model, this curve can be represented by a ray making a certain angle with the bounding line of the half-plane. Show that this angle is equal to Π(δ), the angle of parallelism for the distance δ.

E2751.791.S814.(P.Monsky) Let X be a conic section. Through what points in space do there pass three mutually perpendicular lines, all meeting X ?

E2757.792.(H.D.Ruderman) Let a, b, c be three lines in R3.FindpointsA, B, C on a, b, c respectively such that AB + BC + CA is a minimum.

S12.795.S807.(Klamkin) If a, a1; b, b1; c, c1 denote the lengths of three pairs of opposite sides of an arbitrary tetrahedron, prove that a + a1,b+ b1,c+ c1 satisfy the triangle inequality.

S16.797.(I.J.Schoenberg) Characterize the closed sets S of the complex plane such that d(z + w) ≤ d(z)+d(w) for all complex numbers z and w,whered(z) denotes the euclidean distance from z to S.

E2790.797.S809.(M.D.Meyerson) Suppose we have a collection of squares, one each of 1 area n for n =1, 2, 3,... and any open set G in the plane. Show that we can cover all of G except a set of area 0 by placing some of the squares inside G without overlap. (The edges of the squares are allowed to touch).

S19.798.S812.(Anon, Erewhon-upon-Spanish River∗) Let C be a smooth simpel arc inside the unit disk, except for its endpoints, which are on the boundary. How long must C be if it cuts oﬀ one-third of the disk’s area ? Generalize. ∗ H.Flanders

E2793.798.S819.(E.D.Camier) P and Q are two points isogonally conjugate with respect to a triangle ABC of which the circumcenter, orthocenter, and nine-point center are O, H,and N respectively. If OR = OP + OQ,andU is the point symmetric to R with respect to N,show that the isogonal conjugate of U in the triangle ABC is the intersection V of the lines P1Q and PQ1 where P1 and Q1 are the inverses of P and Q in th circle ABC. (Assume that neither of P , Q is on the circle ABC).

S23.7910.S817.(Garfunkel) Prove that the sum of the distances from the incenter of a triangle to the vertices does not exceed half of the sum of its internal angle bisectors, each extended to its intersection with the circumcircle of the triangle.

E2802.799.S811.(M.Slater) Given a triangle ABC (in the euclidean plane), construct simi- YIU : Problems in Elementary Geometry 116 lar isosceles triangles ABC, ACB outwards on the respective bases AB and AC,andBCA inwards on the base BC (or ABC and ACB inwards and BCA outwards). Show that ABAC (respectively ABAC) is a parallelogram.

E2816.802.S819.(R.Bojanic) Consider a circular segment AOB with AOB < π.LetC be the orthogonal projection of the point B on the line OA. Suppose that the arc AB and the segment CA are each divided into n equal parts. If M is the point of partition of the arc AB closest to B,andN the point of the partition of the segment CA closest to C, show that the projection of the midpoint of the arc MB onto the line OA is always contained in the interval (C, N).

S29.804.S827.(C.Kimberling) Suppose T = ABC is a triangle having sides AB < AC < BC and a point B on segment BC satisfying AB = AB.CallT admissible if the shortest side of triangle T = ABC does not touch the shortest side of T , i.e., theshortestsideofT is BC. (a) Characterize all T for which the sequence T1 = T , T2 = T1, T3 = T2,...consists exclusively of admissible triangles. sn →∞ (b) For such T ,letsn be the length of the shortest side of Tn and determine limn sn+1 . (c*) For such T ,letP be the limit point of the nested triangles Tn and determine the angle AP B.

E2831.805.S819.(M.Cavachi) Prove that a convex hexagon with no side longer than 1 unit must have at least one main diagonal not longer than 2 units.

E2836.807.S825.(J.E.Valentine) Show that an absolute geometry (no parallel postulate) is euclidean (or riemannian) if some triangle has the property that a median and the segment joining the midpoints of the other two sides bisect each other.

E2837.806.S818(C.W.Scherr) Let aij be the side of a triangle that connects vertices i and j.Letmi be the median from vertex i. elementary application of the law of cosines yields the relation t t t t t t t a12a23a31 = λ (a1 + m2 + m3), 4 valid for all triangles when t =2ort =4andλ = 3 .Findanexpressionforλ in the limit as t goes to zero. Find the class of triangles for which the relation is valid for a ﬁxed and arbitrary t.

S34.807.S828.(O.Bottema) In a plane, non-self-intersecting pentagon A1A2A3A4A5 is given. No three of the vertices Ai are collinear and (ijk) denotes the signed area of the oriented triangel AiAjAk.Furthermore,

(124) = a1, (235) = a2, (341) = a3, (452) = a4, (513) = a5. YIU : Problems in Elementary Geometry 117

Determine the area of the pentagon A1A2A3A4A5. The analogous problem, with (123), (234), (345), (451) and (512) being given, was solved by Gauss in 1823. See Crux 3 (1977), p.240.

E2842.807.S8110.(J.Dou) Let T be an isosceles right triangle. Let S bethecirclesuch that the diﬀerence between the areas T ∪ S and T ∩ S is minimal. Show that the center of S divides the altitude on the hypotenuse of T in the golden ratio.

E2843.807.(P.Ungar) A set of√ nonoverlapping rectangles, each having its longer side equal to 1, is inside a circle of diameter 2. Show that the sum of their area is ≤ 1.

E2848.808.S825.(J.Fickett) Prove that the regular tetrahedron has minimum diameter among all tetrahedra that circumscribe a given sphere. (The diameter is the length of a longest edge).

E2866.811.S826.(J.Dou) Let AKL, AMN be equilateral triangles. Prove that the equilateral triangles LMX, NKY are concentric (if Y is on the properly chosen side of NK).

≥ ≤ ◦ E2874.813.S828.(N.Kimura and T.Sekiguchi) Let n 3, 0

E2885.815.S844.(T.Sekiguchi) Let T be a triangle. Construct the set of interior points of T at which the sum of the distances to the sides of T is equal to the arithmetic mean of the lengthes of the altitudes of T .

E2889.815.(I.J.Good) Let P be an arbitrary point in the plane of a regular polygon A1A2 ···An. Let the foot of the perpendicular from P on line AiAi+1 be Qi (where An+1 ± means A1). Let xi be length AiQi: positive if Qi, Ai+1 are on the same side of Ai; negative otherwise. Prove that xi is equal to half the perimeter of the polygon.

E2894.816.(T.Ihringer) Let n be ﬁxed. In how many ways can a square be dissected int (a) n congruent rectangles, (b) n rectangles of equal area ?

E2905.818.S882.(R.J.Stroeker) Inside any triangle ABC,apointP exists such that PAB = PBC = PCA := ω.ThepointP is called a Brocard point and the angle ω is called its Brocard angle. Prove the inequalities 1 1 1 1 3 < + + < ; ω A B C 2ω YIU : Problems in Elementary Geometry 118

3 1 1 1 1 < + + < . 4ω2 A2 B2 C2 ω2

E2906.818.S835.(Garfunkel) Let A,B,C be the intersection of AI, BI, CI with the incircle of ABC. Continue the process by deﬁning I as the incenter of ABC,thenABC (n) (n) (n) π etc. Prove that the angles of A B C approach 3 .

E2911.819.S851.(J.Dou) Let 2 semicircles AC, CB, AC =3CB,begiven.(A, C, B are collinear). Let a abd b be tangentws to the given semicircles at A, B.Letγ be the circle tangent to a and b ant to the larger of the given semicircles. Prove that γ, b and the given semicircles have a common tangent circle. Solution. Invert with respect to the circle with center at the midpoint of AB and diameter equal in length to CB.

ER2914.8110.S857.(R.C.Lyness) AcircleB lies wholly in the interior of a circle A. S is the set of all circles each of which touches B externally and A internally. (i) Find the locus of the internal cneter of similitude of the pairs of circles from S. (ii) Prove that every point of the locus, except one, is teh internal center of similitude of exactly one pair of circles from S.

2917.8110.S846.(F.W.Luttman) Let P0 be a convex polygon of n sides and let 0

E2918.8110.S858.(J.Dou) Show that an isosceles triangle can be dissected symmetrically around the principal median into seven acute isosceles triangles except when the vertex angle is A =90◦, 120◦ or whene 135◦ ≤ A ≤ 144◦.

E2924.821.S857.(Garfunkel) Triangle A1A2A3 is inscribed in a circle; the medians through A1 (A2) meet the circle again at M1 (M2). The angle bisectors through A1 (A2) meet the circle again at T1 (T2). Prove or disprove

|A1M1 − A2M2|≤|A1T1 − A2T2|. YIU : Problems in Elementary Geometry 119

E2930.822.S842.(Monthy Problem Editors) Find the largest square that can be inscribed in some triangle of area 1. See also E3114.

E2950.826.S858.(K.W.Lih) The inner side of a semicircle (including diameter) is a mirror. ≤ ≤ π A light ray emitting from the zenith makes an angle α with the vertical line, 0 α 2 . Characterize α such that the light ray will hit the zenith after ﬁnitely many reﬂections.

E2958.827.S854.(Klamkin) Let x, y, z be positive, and let A, B, C be angles of a triangle. Prove that x2 + y2 + z2 ≥ 2yz sin(A − 30◦)+2zxsin(B − 30◦)+2xy sin(C − 30◦).

E2959.827.S855.(Garfunkel) Triangle ABC is inscribed in a circle. The medians of the triangle intersect at G and are extended to the circle to points D, E, F.ProvethatAG + BG+ CG ≤ GD + GE + GF . This is equivalent to part (a) of E2505.

E2962.828.S854.(Klamkin) It is known that if the circumradii R of the four faces of a tetrahedron are congruent, then the four facees of the tetrahedron are mutually congruent (i.e., the tetrahedron is isosceles. (See, for example, Crux Math. 6 (1980) 219). It is also known that if the inradii r of the four faces of a tetrahedron are congruent, then the tetrahedron need not be isosceles. (See, for example, Crux Math. 4 (1978) 263). Show that if Rr is the same for each face of a tetrahedron, the tetrahedron is isosceles.

E2963.828.(C.P.Popescu) Let A1A2A3, A1A2A3 be two equilateral triangles in the plane. − Construct circles γi, γi with radii ri (ri) and centers A i (Ai), i =1, 2, 3. Suppose further that ri (ri) are geometric progressions with ratio a positive integer. When can the six circles be concurrent ?

E2966.828.S888.(P.J.Giblin) A, B, P1,P2,P3 are distinct points in the plane. PiPjA, PiPjB are proper triangles, i.e.,notwoofP1,P2,P3 are collinear with A or with B.The anticlockwise angles from AP1 to AP2, AP1 to AP3, BP1 to BP2, BP1 to BP3 are θ1,θ2,φ1,φ2. If ai = APi, bi = BPi, and if the relations sin θ a sin θ a sin(θ − θ ) a 1 3 = 2 2 = 2 1 1 sin φ1 b3 sin φ2 b2 sin(φ2 − φ1) b1 hold, show that the angle APiB has the same pair of bisectors as one of the angles of the triangle P1P2P3. (Possibly the internal bisector of one angle is the external bisector of the other).

E2967.828.(J.Dou) Divide a circle into four equiareal parts with (i) arcs (ii) segments of minimal total length. YIU : Problems in Elementary Geometry 120

E2968.829.S855.(G.Tsintsifas) The points A1,A2,A3 lie on the sides A2A3, A3A1, A1A2 of an acute angle triangle A1A2A3 respectively. Show that ≥ 2 ai cos Ai ai cos Ai

where a1,a2,a3 are the sides of the triangle A1A2A3 and a1,a2,a3 are the sides of the triangle A1A2A3.

E2974.8210,(Correction 837).S856.(J.Dou) Let AMB (oriented clockwise) and CMD (counterclockwise) be similar triangles. Prove that triangles ACX (clockwise) and YDB(counterclockwise), both similar to the ﬁrst triangles, have the same circumcenter.

E2980.831.S926.(J.Dou) Given the points A1,A2,A3,M and the line s,constructPQ such that PQ is equal and parallel to A1M and P1Q1 = P2Q2 = P3Q3,wherePi, Qi are the intersections of PAi, QAi with s. Describe the locus of the point M for which the problem has a solution when A1, A2, A3 and s are known, (ﬁxed).

E2981.831.S864.(Klamkin) If the three medians of a spherical triangle are equal, must the triangle be equilateral? Note that the sides of a proper spherical triangle are minor arc of great circles and thus its perimeter is < 2π.

E2983.831.S897.(E,Ehrhart) Let ABC be an equilateral triangle of perimeter 3a.Calcu- late the area of the convex region consisting of all points P such that PA+ PB+ PC ≤ 2a.

n E2987.832.S859.(G.Tsintsifas) Let Sn = A1A2 ···An+1 be an n−simplex in R and M a point insides its circumsphere S :(0,R). The straight line AiM intersects the sphere (0,R)at n+1 AiM the point A .WedenoteK = .LetG be the centroid of Sn.Prove i i=1 MAi (a) K>n+1ifandonlyifM lies outside the sphere (s) with diameter OG. (b) K = n +1ifandonlyifM lies on the sphere (s). (c) K

E2990.833.S862.(H.Eves and C.Kimberling) Let ABC be a triangle and L a line in the plane of ABC not passing through A, B, C. (a) Prove that the isogonal conjugate of L is an ellipse, parabola or hyperbola according as L meets the circumcircle of ABC in zero, one or two points. (b) Prove that the isotomic conjugate of L is an ellipse, parabola or hyperbola according as L meets E in zero, one or two points, where E is the ellipse through A, B, C having the centroid of triangle ABC as center.

E2992.834.(J.Dou) Find teh shape of a contour of length L that encloses the largest possible YIU : Problems in Elementary Geometry 121 area and is constrained to pass through three given points.

E2997.835.S864.(I.Adler) Let p0 be the perimeter of an inscribed regular n-gon in a unit circle, and let dk be the distance from the center of the circle to the side of the inscribed regular 2kn-gon. Prove that ∞ p 1 0 = π. 2 d k=1 k

E3007.837.S867.(G.Odom) Let A and B be the midpoints of the sides EF and ED of an equilateral triangle DEF.ExtendAB to meet the circumcircle (of DEF at C. Show that B divides AC according to the golden section. Solution without words.

E3009.837.S8610.(C.Jantzen) Points X, Y, Z are chosen on the sides of ABC such that AX BY CZ = = = k XB YC ZA and a triangle PQR is formed using CX,AY,BZ as sides. The operation is repeated on PQR, that is the points X.Y ,Z are chosen on the sides of PQR such that PX QY RZ = = = k XQ Y R ZP and a triangle LMN is formed using RX,PY,QZ as sides. Show that LMN is similar to ABC and ﬁnd the ratio of similarity.

E3013.838.S869.(S.Rabinowitz) Let ABC be a ﬁxed triangle in the plane. Let T be the transformation of the plane that maps a point P into its isotomic conjgatte (relative to ABC). Let G be the transformation that maps P into its isogonal conjugate. Prove that the mappings TG and GT are aﬃne collineations (linear transformations).

E3020.839.S868.(C.Kimberling) Suppose ABC is a nonisosceles triangle. Find three hyperbolas concurrent in a point P such that triangles AP B, AP C, BP C all have the same perimeter. How does this common perimeter compare with that of ABC?

E3038.842.S869.(T.Sekiguchi)√ Prove the inequalities. ≤ 3 3 (a) sin A +sinB +sinC√ 2 . ≤ 3 3 (b) sin A sin B sin C 8 .

E3044.845.S871.(J.Dou) Construct ABC given r, AI, AH. YIU : Problems in Elementary Geometry 122

E3045.845.S874.(C.P.Poposcu) Let H be a hexagon inscribed in a circle. Show that H can be circumscribed about a conic if and only if the product of three alternate sides equals the product of the other three.

E3049.847.S882.(J.Dou) Determine a planar region of area 4 which can be partitioned into four subregions of unit area in such a way that the total length of all bounding arcs is a minimum. For example, a square of area 4 can be partitioned into four unit squares so that the bounding arcs have total length 12, while a circle of area 4 can be partitioned into four sectors 4√π+8 ≈ of unit area so that the total length of the bounding arcs is π 11.6.

E3050.847.(J.Dou) A optimal tripartition of a plane region is a partition into three subregions of equal area such that the total length of the separating arcs is a minimum. Determine those isosceles triangles which admits (a) two optimal tripartitions, (b) two optimal tripartition among those partitions using only straight line segments.

E3051.847.(P.O’Hara and H.Sherwood) It is observed in plane analytic geometry that any set S that is symmetric with respect to both the x and y axes is also symmetric with respect to the origin. Does the statement remain valid if the y−axis is replaced by a line through the origin with inclination angle α ?

E3054.848.S8610.(V.D.Mascioni) Prove the inequalities. ≥ 2π 2 (a) abcABC ( 3 ) r . √ − − − ≥ 4π 3 3 (b) abc(π A)(π B)(π C) ( 9 ) s .

E3059.849.(M.Guan and W.Li) Let H be a regular n-gon with side length equal to one, n ≥ 4. Show that if K is any n-gon inscribed in H with side-length xi,i=1, 2,...,n,then n − ≤ 2 ≤ − n(1 cos θ)/2 xi n(1 cos θ), i=1 where θ is the internal angle of H. Discuss the case when equality holds.

E3068.8410.S872.(Tsintsifas)) Let T = ABC be a triangle with inradius r and circumradius R. We consider a circular disc C with radius d, r ≤ d ≤ R, in a position such that Area(T ∩ C) is a maximum. Prove that as d varies continuously in the closed interval [r, R], the center of the disc C (in the maximum position) moves on a conic τ passing through the incenter, circumcenter and the Lemoine point of ABC.Also,ABC is self-polar with respect to the conic τ.

E3071.851.S882.(K.Satyanarayana) In ABC, A < B < C.ProvethatI lies inside YIU : Problems in Elementary Geometry 123

OBH.

E3073.852.S882.(C.P.Popescu) In the hexagon A1A2A3A4A5A6, the triangles A1A3A5 and A2A4A6 are equilateral. Is it true that AkAk+3 are congruent if their sum equals the perimeter of the hexagon? Also, consider the converse.

E3080.853.(L.C.Larson) Can the following equations be satisﬁed with integers?

(x +1)2 + a2 =(x +2)2 + b2 =(x +3)2 + c2 =(x +4)2 + d2.

E3084.854.S883.(P.Pamﬁlos) Given a family of circles in the plane all of which passes through a common point and no two of which are equal, show that there is another circle enveloping all the circles of the family if and only if there is a straight line containing all the intersection points of the common tangents of any two circles of the family.

E3091.855.S875.(C.P.Popescu) Equilateral triangle ABC is inscribed in XY Z,withA between Y and Z, B between Z and X,andC between X and Y . Show that

XA + YB+ ZC < XY + YZ+ ZX.

Is equality possible?

E3098.856.S876.(R.Cuculière) Given two circles with diameters IA = a, IB = b,anda set of smaller circles between them as in the following figures, find the total area enclosed by the small shaded circles in each of the folowing cases: (a) The center of one of the small circles lies on the common diameter of the larger circles. (b) Two of the small circles are tangent to the diameter of the large circles. (c) No restrictions.

E3114.859.S881.(M.J.Pelling) Find the largest cube that can be inscribed in some tetrahedron of volume 1. See also E2930. YIU : Problems in Elementary Geometry 124

E3134.862.S885.(J.Dou) Construct ABC given ma,wa,A.

E3135.863.S888.(H.Demir) For a scalene triangle ABC inscribed in a circle, prove that there is a point D on the arc of the circle opposite to some vertex whose distance from this vertex is the sum of its distances from the other two vertices. Show how D may be constructed with straightedge and compass.

E3146.864.S888.(J.J.Wahl) Prove or disprove: √ √ √ 2s( s − a + s − b + s − c) ≤ 3( bc(s − a)+ ca(s − b)+ ab(s − c)).

The statement is correct.

E3150.865.S887,9010.(G.A.Tsintsifas) For arbitrary positive real numbers p, q, r, p q r √ a2 + b2 + c2 ≥ 2 3. q + r r + p p + q

E3154.866.S887.(G.A.Tsintsifas) Let A1,B1,C1 be points on the sides a, b, c of ABC respectively, and a1,b1,c1 the sides of A1B1C1.Provethat

2 2 2 2 a b1c1 + b c1a1 + c a1b1 ≥ 4 .

E3155.866.S889.(G.Bennet, J.Glenn and C.Kimberling) Prove that for any ABC, there exist points A,B,C such that (1) A lies on BC, B on AC and C on AB. (2) AC + CB = BA + AC = CB + BA. (3) AA,BB,CC concur in a point.

E3157.866.S886.(L.I.Nicolaescu) How many sets of four distinct points forming the vertices of a trapezoid are there if the points are chosen from the vertices of a regular n−gon, n ≥ ?

E3164.867.S887.(H.Demir) Let s, t be the lengths of the tangent line segments to an ellipse s from an exterior point. Find the extreme values of the ratio t .

E3167.868.(E.B.Cossi) Let Pi be any one the ﬁve regular polyhedra inscribed in a unit sphere. For each polyhedron Pi, determine the smallest and the largest number of vertices of Pi which can be seen from a point on a concentric sphere of radius R>1. YIU : Problems in Elementary Geometry 125

E3172.869.S8810.8910.(J.Dou) Let A (respectively B, C be the foot of the altitude from vertex A (respectively B, C), in a triangle ABC.LetH be its orthocenter, adn M be an arbitrary point of the plane. Prove that the conics MABAB, MBCBC, MCACA, MHCAB, MHABC and MHBCA have a common point other thatn M.

E3177.8610.S897.(J.Dou) Let A, B, C be three points on a circle. Let A1 be the intersection of the tangent at A with the line through BC, similarly for B1,C1. Prove that the circles ABB1,BCC1,CAA1 and the line A1B1C1 have a common point.

E3180.8610.S888.(Klamkin) Prove that

A B C A B C cos +cos cos 1+sin +sin sin . 2 2 ≥ 2 2 2

E3183.871.S892.(Klamkin) Let P denote the convex n-gon whose vertices are the midpoints of the sides of a given convex n-gon P . Determine the extreme values of (i) Area P /Area P , (ii) Perimeter P / Perimeter P .

E3185.871.(R.E.Spaulding) Let P be a point in the interior of an equilateral triangle, and let S be the sum of the perpendicular distances to the three sides of the triangle from P .In euclidean geometry, the sum S always equals the altitude of the triangle. In Lobachevskian geometry, prove that S is less than any altitude. In addition, ﬁnd the position of P which would give a minimum value for S.

E3193.872.S898.(A.Lenard) Let θ be an undirected acute angle. Show that if a one-to-one mapping T of the euclidean plane E onto itself has the property that whenever points P and Q subtend the angle θ at the point R then also the points T (P )andT (Q) subtend the angle θ at the point T (R), then T is a similarity transformation of E.

E3195.873.S896.(L.Kuipers) Given ABC, consider those inscribed ellipses touching AB in C1, BC in A1 and CA in B1 with AB1.BA1 = B1C.A1C. Describe the locus of the centers of such ellipses.

E3199.873.S897.(Guelicher) In the triangle ABC,pointQ is on the ray BA,pointR is on the ray CB,andBQ = CR = AC. A line parallel to AC through R intersects CQ in a point T . A line parallel to BC through T intersects AC in a point S. Show that

(AC)3 = AQ · BC · CS. YIU : Problems in Elementary Geometry 126

E3208.873.S891.() Suppose that the euclidean plane, line segments of lengths a, b, c, d emanate from a given point P in clockwise order, where a, b, c, d are given positive numbers with a2 + c2 = b2 + d2. (i) Show that the four segments can be so placed that the endpoints determine a rectangle containing P , and show that this rectangle may have any speciﬁed area between 0 and some maximum value M(a, b, c, d). (ii) Determine M(a, b, c, d). See also MG1147.

E3209.873.S893 Family of circles.

E3231.879.S895.(H.Guelicher) In a triangle P1P2P3 let pi bethesideoppositevertexPi and let si be a line parallel to pi (but diﬀerent from p−I). Suppose that si divides PiPi+1 in the PiQi − signed ration λi so that if si meets pi 1 in Qi,thenλi = QiPi+1 . Prove that the lines s1,s2,s3 are concurrent if and only if λ1λ2λ3 − (λ1 + λ2 + λ3)=2.

E3232.879.S896.(J.Dou) Given lines li,1≤ i ≤ 5andpointsQi,1≤ i ≤ 5, in the plane such that Qi does not lie on li, prove that there exist points Pi, Ri on line li such that the angle PiQiRi is a right angle and such that the ten points Pi, Ri, i =1, 2,...,5 lie on a conic.

E3236.879.S906.(N.D.Rlkies) For a plane triangle call two circles within the triangle companion incircles if they are the incircles of the two triangles formed by dividing the given triangle by passing a line through one of the vertices and some point on the opposite side. (a) Show that any chain of circles C1,C2,... such that Ci and Ci+1 are companion incircles for every i consists of at most six distinct circles. (b) Give a ruler and compass construction for the unique chain which has three distinct circles. (c) For such a chain of three circles show that the three subdividing lines are concurrent.

E3239.8710.(Klamkin) Show that if A is any three dimensional vector and B, C are unit vectors, then [(A + B) × (A + C)] × (B × C) · (B + C)=0. Interpret the result as a property of spherical triangles.

E3251.882.S907.(C.Kimberling) Inside a given ABC, it is possible to construct three circles each touching the other externally and one side of the triangle. Let D, E, F be the points of contact of these circles, appropriately labelled. YIU : Problems in Elementary Geometry 127

(a) Prove that the lines AD, BE, CF are concurrent. (b) Prove that the lines IaD, IbE,IcF are concurrent.

E3256.883.(J.Isbell) (a) Let T be the set of triangles in the plane whose vertices have integral coordinates and whose sides have integral lengths. Certain isosceles triangles in T can be constructed by ﬁtting together two congruent right triangles in T , e.g. the isosceles triangle with vertices (−12, −9), (12, 9), (−12, 16) arises in this way. Are there any other isosceles triangles in T ? (b) Consider the set V of triangles in 3-sapce whose vertices have integral coordinates. Does V contain any equilateral triangles with integral side-length ?

E3257.883.S901.(I.A.Sakmar) Let P, Q, R be the new vertices of equilateral triangles constructed outwardly on the edges of a given triangle ABC. (a) Show that any triangle PQR which can be obtained in this way arises from a unique triangle ABC, and give a construction for recovering triangle ABC from triangle PQR. (b) Show tht not every triangle PQR can be so obtained.

E3259.884.S902.(J.Dou) Let R be a semicircular region bounded by a line L and a semicircle S with center on L. Suppose P1 and P2 are given points in the interior of R.Wewishto ﬁnd parallel lines l1,l2 through P1, P2 such that P C P C 1 1 = 2 2 , P1D1 P2D2 where C1,D1 are the intersections of l1 with S and L and C2, D2 are the intersection of l2 with S and L. Give a necessary and suﬃcient condition on P1,P2 for such parallel lines to exist.

E3270.886.S899.(B.Lindstr¨√ om) Determine those positive rational numbers m for which arctan m is a rational multiple of π.

E3279.887.() For some n ﬁnd a simplex in En with integer edges and volume 1. See also MG871.1261.S882

E3282.888.S9010.(D.M.Milosevic) Prove the inequality R w2 + w2 + w2 ≤ s2 − r( − r), a b c 2 with equality if and only if the triangle is equilateral.

E3283.888.(O.Frink) Every simple closed polygon in the plane has three centroids, namely the centroid of its vertex set, the centroid of its boundary, and the centroid of its interior. In general, all three are distinct. YIU : Problems in Elementary Geometry 128

(a) In the case of a triangle show that these centroids coincide if and only if the triangle is equilateral. (b) Which of the three centroids are aﬃne invariant ?

E3293.889.S911.(J.Keane and G.Patruno) Suppose that the distinct circles C1,C2 intersect at P and Q. Suppose that the tangent to C1 at P intersects C2 again at A, the tangent to C2 at P intersects C1 again at B, and the line AB separates P and Q.LetC3 be the circle externally tangent to C1, externally tangent to C2, tangent to line AB, and lying on the same side of AB as Q. Prove that the circles C1 and C2 intercept equal segments on one of the tangents ot C3 through P .

E3299.8810.(J.Yamout) Suppose ABCD is a plane quadrilateral with no two sides parallel. Let AB and CD intersects at E and AD, BC intersect at F .IfM,N,P are the midpoints of AC, BD, EF respectively, and AE = a · AB, AF = b · AD,wherea and b are nonzero real numbers, prove that MP = ab · MN.

E3307.892.S918.() The celebrated Morley triangle of a given triangle ABC is the equilateral triangle whose vertices are the intersections of adjacent pairs of internal angles trisectors of ABC. Determine the maximum values of (i) sM /s, (ii) RM /R, (iii) rM /r, (iv) M /.

E3314.893.S909.() Let P be a point inside acute triangle ABC.Putα1 = PAC,β1 = PBA,γ1 = PCB.Provethat

3 1 1 cot α1 +cotβ1 +cotγ1 > 2 2 3 4 (cot B +cotB +cotC) 2 .

E3369.902.() Suppose we are given a piece of paper in the shape of an equilateral triangle. Suppose P is a point in the intersection of the three open circular discs with diameters AB, BC, CA. If we fold the three corners of the paper in such a way that the vertices coincide with P , we get a hexagon three sides of which are the creases formed and the other three sides YIU : Problems in Elementary Geometry 129 of which are portions of the sides of ABC. Prove that the area and perimeter of this hexagon are both maximized when P is the centroid of ABC.

E3375.903 Express ∞ 1 tan arctan 2 n=1 n in closed form.

E3377.903.S918.() Suppose we consider the polygon with vertices e2πiθj ,j =1, 2,...,n in the complex plane, with 0 ≤ θ1 <θ2 < ···<θn < 2π. (0) ··· Suppose α and β are given positive numbers with α + β = 1. Deﬁne θj = θj for j =1, 2, ,n and (k) (k−1) (k−1) θj = αθj + βθj+1 for k =1, 2, ... where subscripts are taken modulo n and the values are taken modulo 2π.Prove that (k) − (k) 2π lim (θj+1 θj )= , k→∞ n for j =1, 2, ···.

E3392.906.S922.(A.Bege) Given an acute triangle ABC with orthocenter H,letA1,B1,C1 be the feet of the altitudes from A, B, C respectively, and let A2,B2,C2 be the feet of the perpendicualars from H onto B1C1,C1A1,A1B1 respectively. Prove that

ABC ≥ 16A2B2C2, and determine when equality holds.

E3397.907.S921.(J.Chen and C.H.Lo) The perimeter of ABC is divided into three equal parts by three points P, Q, R. Show that 2 PQR> ABC, 9

2 and that the constant 9 is best possible.

E3407.909.S934.(C.Kimberling) Suppose ABC is a given triangle. Prove the existence of a triangle that is in perspective with every antipedal triangle of ABC. YIU : Problems in Elementary Geometry 130

E3408.909.S922.(J.V.Savall and J.Ferrer) For each positive integer k,letf(k)denote the number of triangles with integer sides and area k times the perimeter. It is well known (cf. E2420.73?.S74?) that f(1) = 5. Obtain an upper bound for f(k)intermsofk. The analogous problem for right triangle appeared in CM1447.15?, CMJ232.825.S843.

E3414.9010.S923.(G.Myerson) Suppose we construct a sequence of rectangles as follows. We begin with a square of area one. We then alternate adjoining a rectangle of area one alongside or on topof the previous rectangle. Find the limiting ratio of length to height. π Answer. 2 . Beautiful solution by R.M.Robinson.

E3417.911.S927.(R.S.Luthar) Suppose ABC is a triangle with AB = AC,andletD, E, F, G be points on the line through B and C deﬁned as follows: D is the midpoint of BNC, AE is the bisector of BAC, F is the foot of the perpeandicualr from A to BC,andAG is perpendicular to AE (i.e. AG bisects one of the exterior angles at A). Prove that AB · AC = DF · EG. Solution. Assume b>c. EG = EB + BG ac ac = + b + c b − c 2abc = b2 − c2 2abc = CF2 − BF2 2abc = (CF + BF)(CF − BF) 2abc = 2a · DF bc = . DF

E3422.912.S927.(H.Demir and C.Tezer) Suppose F and F are points situated symmetrically with respect to the center of a given circle, and suppose S is a point on th circle not on the line FF.LetP and P be the second points of intersection of SF and SF respectively with the circle. If the tangents to the circle at P and P intersect at T , prove that the perpendicualr bisector of FF passes through the midpoint of the line segment ST.

E3434.914.(John P. Hoyt) Prove or disprove that there are inﬁnitely many triples of positive integers (a, b, c) with no common factor such that the triangle ABC with sides a, b, c has the following property: the median from A, the angle bisector from B and the altitude from C are concurrent. Examples of such triples are (12, 13, 15), (35, 277, 308), (26598, 26447, 3193). YIU : Problems in Elementary Geometry 131

See R.K.Guy, My favorite elliptic curve: A tale of two types of triangles, Amer. Math. Monthly, 102 (1995) 771 – 781. See also E374.393.S403, where the printed solution is not correct.

E3438.91?S929.(H.G¨licher) Let P1P2P3 have the longest side P1P2. For each of the six permutations of 1,2,3, let Pij be the point on the ray PiPj such that PkPiPij = PiPjPk.Let pij be the length of PkPij and let pi be the length of PjPk.Provethat 2 2 2 p12 p21 (i) p1 + p2 = p3 if and only if p13 + p23 =1; 3 3 3 p31 p32 (ii) p1 + p2 = p3 if and only if p13 + p23 =1.

E3443.915.S928.(C.P.Popescu) Let Ai,i =0, 1,...,5 denote the vertices of a hexagon inscribed in a circle and let Bi denote the intersection of the straight lines AiAi+2 and Ai+1Ai+3 for i =0, 1,...,5, the indices being computed modulo 6. Prove that if the triangles A0A2A4 and A1A3A5 have the same orthocenter, then the straight lines BiBi+3,i=0, 1, 2 are concurrent.

E3450.916.S933.(D.Bowman) Let T (n) be the number of triangles lying in the subset [0,n]×[0,n] of the plane whose sides lie on lines of slope 0, ∞, 1, −1 passing through points with integer coordinates. Derive a closed formula for T (n).

E3460.918.S931.(E.Ehrhart) (a) Suppose we have n mutually perpendicualr chords through apointinteriortoasphereS in n−dimensional euclidean space. Prove that the sum of the squares of the lengths of these chords depends only on the radius r of the sphere and the distance d from P to the center of the sphere. (b) More generally, suppose 1 ≤ k ≤ n. Each set of k of the n mutually perpendicular chords through P given in (a) determines a k−dimensional aﬃne subspace. Prove that the sum of the n 2 −powers of the k−dimensional measure of the cross sections of S made by these aﬃne k k subspaces depends only on r and d.

E3466.919.S934.(W.Fenton) Suppose ABC is given. If X is a point not on any of the lines BC, CA, AB, let the lines AX, BX, CX meet these lines respectively in points A,B,C. It is known (Miquel’s theorem) that the circles ABC,ABC,ABC intersect in a point Y . Prove that X = Y if and only if X is the orthocenter of ABC.

E3468.919.() Suppose m and n are positive integers such that all primes factors of n are larger than m. (a) Prove that n kπ φ(n) − µ(n) 2m ∗ sin2m( )= , n 4m m k=1 YIU : Problems in Elementary Geometry 132 where * denotes summation over integers relatively prime to n. (b) Find a similar formula for cosines.

E3469.9110.S939.(H.Demir) Suppose P is a point in the interior of ABC,andAP, BP, CP meet the lines BC, CA, AB respectively at the points D, E, F. Prove that the centroids of the six triangles PBD,PDC,PCE,PEA,PAF,PFB lie on a conic if and only if P lies on at least one of the three medians of the triangle. o YIU : Problems in Elementary Geometry 133

American Mathematical Monthly Advanced Problems, 1976 – 1995.

AMM5499.67?.S68(p.1019),801.(D.E.Daykin) (1) Are there rational numbers a and b such that no rational number c exist for which there is a triangle having sides a, b, c and area k ? (2) Prove or disprove the conjecture: for every positive rational number k for which there exist positive rational a, b, c such that a triangle with sides a, b, c has area k.

AMM5790.71?.S823.(D.E.Daykin) Find all nontrivial maps f : R2 → R2 such that whenever a, b, c are collinear, then f(a),f(b),f(c) are collinear.

AMM5986.74?.S764.(D.E.Daykin) Let E be the real euclidean plane and 0 <α<1. What can be said about maps f : E → E which send each triangle T into a triangle fT with area fT ≤ α area T ?

AMM5973.74?S762.(G.Tsintsifas) Let G = {A1,A2,...,An} be a bounded set of points in the plane. If any three of these points can be covered by a strip of breadth d,showthtG can be covered by a stripof breadth 2 d. Find also the minimum real number k, so that any point set G with the given property can be covered by a stripof breadth kd.

AMM6062.75?.S777,812,827.(B.H.Voorhees) Consider an inﬁnite sequence of regular n−gons such that each(n +1)− gon is contained within the preceding n−gon and is of maximal area consistent with this constraint. Take the ﬁrst element of this sequence as an equilateral triangle having unit area. Is the limit of this sequence a point or a circle ? If it is a circle, determine its area.

AMM6089.764.(E.Ehrhart) Let K be a convex body in Rn of Jordan content V (K) > n (n+1) ∪ − n! , and with centroid at the origin. Does K ( K) contain a convex body C, symmetric in the origin, for which V (C) > 2n ?

AMM6178.779.S795.(R.Kowalski) Deﬁne the shape of a rectangle to be the ratio of the longer side to the shorter side. Suppose one has an unlimited number of congruent squares at one’s disposal. Given shape α andanerrorJ, what is the least number of squares one needs to construct a rectangle whose shape diﬀers from α by less than J ? YIU : Problems in Elementary Geometry 134

AMM6179.779.S8010.(E.Ehrhart) Find all cubes in a cubic lattice whose vertices are lattice points.

AMM6223.787.S799.(H.D.Ruderman) Let C be a convex curve. Let Q be a convex curve such that the two tangents to C from each point P of Q form an angle θ ﬁxed in size. Assume that all points are in the same plane. (1) If θ =90◦ and Q is a circle, must C be a circle or an ellipse ? (2) If C is an ellipse and θ =90 ◦, what is the nature of Q ?

AMM6298.805.S824.(J.L.Brenner) If an arbitrary set of 19 lattice points (with integer coordinates) is given in euclidean 3-space, prove that some three have a centroid with integer coordinates. (This assertion is false if 19 is replaced by 18).

AMM6316.809.S829.(D.Winter) Let S be a set of 3n points in R3, no four of which are coplanar. Suppose that S = R ∪ Y ∪ G,whereeachofR, Y , G has n points. Is it possible to partition S into n triples {ri,yi,gi},1≤ i ≤ n,whereeachri is in R,eachyi is in Y ,andeach gi is in G, in such a way that the n triangles Ti = conv{ri,yi,gi} are pairwise disjoint ?

AMM6322.8010.(M.J.Pelling and Erd¨os) Find the largest constant k such that there exists a set S of planar measure k, no three points of which form the vertices of a triangle of − 3 area 1. In particular, is k =4π3 2 ?

AMM6364.819.S835.(K.B.Lsisenring) A circle with center at the vertex and radius equal to the latus rectum meets a parabola at P, Q. The circle and parabola have common tangents meeting the parabola at X, Y .ProvethatXP,Y Q are tangent to the circle.

AMM6367.819.S838.(A.Ehrenfeucht and J,Mycielski) Let A be a ﬁnite collection of distinct but possibly overlapping regular n−gons of the same size on the plane such that every vertex of every n−gon of A is a vertex of exactly two n−gons of A. (a) Construct a collection A of 2nn−gons such that, even if the n−gons are rigid, A is ﬂexible. (b) For which n is rigid A possible ?

AMM6377.822.(K.R.Kellum) Suppose G is a subset of th euclidean plane such that G meets each vertical line in exactly two points and G meets each nonvertical line in a dense set of points. Must G have a subset H such that H meets each vertical line in one point and each nonvertical line in a dense set of points ?

AMM6381.823.(W.W.Meyer) M.D.Fox (AMM 87 (1980) 708 – 715) defines a Steiner YIU : Problems in Elementary Geometry 135 chain as a sequence of circles each touching its two neighbours and two given boundary circles CO and CI . Enlarging on this, with the hypothesis that CO suurounds CI , we define a linear chain as a polygon circumscribed by Co and circumscribing CI . Linear or circular, a chain is n said to have period m if it closes on itself, the first link and the (n+1)th link coinciding, after m cycles arounds C0. Proe that a linear chain of rational period p and a circular chain of rational period q coexists if and only if π π π π p>2,q>2, cos cos( − ) ≤ cos 2p 4 2q 4 and then the boundary circles are uniquely determined as to relative size and eccentricity.

AMM6385.824,S839.(M.D.Meyershon) Prove or disprove: every simple closed curve in euclidean space contains the vertices of a rectangle. (It is known to be true in the euclidean plane).

AMM6388.825.S8310.(N.Wheeler and H.Straubing) A regular tetrahedron R sits on a unit triangle T on a plane tiled with triangles congruent to T . A move consists in rotating R about an edge in contact with the plane. After several moves, R sits on T again. Have the vertices of R been permuted in space ? What if R is cube and the tiling is by squares.

AMM6418.831.S848.(G.Benke) Prove that

2N−1 πn2 sin 2N πn = N. n=1 sin 2N Original published solution is incorrect.

AMM6477.849.S865;8910. (L.Funar) Let r be the radius of√ the incircle of an arbitrary ≤ 5−1 triangle lying in the closed unit square. Prove or disprove that r 4 . Original published solution is incorrect.

AMM6478.849.(L.Funar) Let r be the radius of the incircle of an arbitrary triangle lying in a closed ﬁgure F of width w,andletR be the radius of the incircle of F . Are the following inequalities valid ? 1 ≤ sup r ≤ 1 (a) 4 w 2 ; 1 ≤ sup r ≤ (ii) 2 R 1.

AMM6557.879.S906.(C.Kimberling) Let C be the circumcircle of ABC.LetA be the point, other than A,wheretheA-median of ABC meets C.LetA,bethepoint,otherthan A,wheretheA-altitude of ABC meets C. Similarly deﬁne B,C nd B,C.LetDEF be the YIU : Problems in Elementary Geometry 136 tangential triangle of ABC (D is the point where the line tangent to C meets the line tangent to C at C). Prove that the lines DA,EB,FC and the lines DA,EB,FC concur in points that lie on the Euler line of ABC. Answer: (a2(b4 + c4 − a4):···: ···) and (tan A cos 2A : ···: ···).

AMM6560.8710.(A.J.Krishna,M.M.Rao, and G.S.Rao) If x and y are odd positive integers, evaluate ∞ 1 nπ nπ 2 tan tan . n=1 n x y

AMM6571.88?.S917.() Let A(n) be the maximum area of a polygon of n sides of lengths 1, 2,...,n,wheren ≥ 4. It is known that the maximum area occurs for a polygon inscribed in a circle. (cf. Polya, Mathematics and Plausible Reasoning, vol.1, pp.174-177). Let B(n)denote the area of a regular polygon with n sides and perimeter 1 + 2 + ···+ n.Provethat A(n) π2 1 − . B(n) 3n2

AMM6605.896.(E.Ehrhart) If k is a poisitive integer, Schinzel, Enseignement Math. (2) 4 (1958) 71 – 72, proved that the circle 1 2 5k 2 X − + Y 2 = 3 3 passes through exactly 2k + 1 lattices points; clearly, the two coordinates of any one of these 2k + 1 lattice points are of like parity. Thus, by making the substitution X = x + y, Y = x − y, we see that the smaller circle √ 1 2 1 2 5k 2 2 x − + y − = (∗) 6 6 6 also passes through exactly 2k + 1 lattice points. (i) Show that when k = 1 no circle smaller than (*) passes through exactly 3 lattice points. (ii) Show that when k = 2 no circle smaller than (*) passes through exactly 5 lattice points. (iii) Show that when 2k + 1 is composite, there is a circle smaller than (*) which passes through exactly 2k + 1 lattice points.

AMM6628.904.S918.() Call a triangle a Heron triangle if it has integer sides and integer area. Fermat showed that there does not exist a Heron right triangle whose area is a perfect square. However, the triangle with sides 9, 10, 17 has area 36. Prove that there are inﬁnitely many Heron triangles whose sides have no common factor and whose area is a perfect square. YIU : Problems in Elementary Geometry 137

Solution. (C.R.Maderer) For each positive integer k, deﬁne

a(k):=20k4 +4k2 +1, b(k):=8k6 − 4k4 − 2k2 +1, c(k):=8k6 +8k4 +10k2.

Here, a(k),b(k)

[(2k)(2k2 − 1)(2k2 +1)]2.

See also N.J.Fine, On rational triangles, AMM 83 (1976) 517 – 521. (J.Buddenhagen) There are inﬁnitely many pairs of Heron triangles which share the same 1 2 − square area. Let m>1 be an odd integer such that 2 (m 1) is a square. The triangles with sides 1 1 (m3 + m2) − 1, (m3 − m2)+1,m2; 2 2 1 m +1 m3 − (m − 1),m3 − ,m 2 2 1 2 2 − both have area 2 m (m 1). A triangle has rational area if and only if the numbers t = ,t= ,t= , 1 s(s − a) 2 s(s − b) 3 s(s − c)

1−t1t2 are rational. Note that t3 = t1+t2 . The area of the triangle is a rational square if and only if

2 u = t1t2(t1 + t2)(1 − t1t2).

1 The conclusion follows from the observation that elliptic curve for t2 = 4 has positive rank. (Editor’s comment): N.Elkies, On A4 + B4 + C4 = D4, Math. Comp/ 51 (1988) 825 – 835, has proved that this equation has inﬁnitely many solutions with gcd(A, B, C, D) = 1. Since fourth powers are congruent to 1 or 0 modulo 16, in such a quadruple, exactly one of A, B, C is odd. If we choose a = B4 + C4, b = A4 + C4,andc = A4 + B4,thengcd(a, b, c) = 1, and a, b, c form a triangle with are (ABCD)2. YIU : Problems in Elementary Geometry 138

American Mathematical Monthly Elementary Problems, 1976 – 1991.

AMM.741,p61.() A generalization of Morley’s theorem. Coxeter-Greitzer p.49,163: Morley’s theorem as a consequence of the fact that A1A2, B1B2 and C1C2 are concurrent

E1822.65?.S777.(N.Ucoluk) Let A, A1 and B,B1 be any two pairs of points in the plane. Consider the locus of points N such that the angles ANA1 and BNB1 (with measures having absolute values α and β respectively) satisfy the codition α = kβ,wherek is a given positive real number. (a) Determine the diﬀerentiability properties of this locus, and (b) when the tangent line exists give a geometric procedure (ﬁnite) for its construction.

E2319.72?.S761.(C.S.Ogilvy) Find the side of the largest cube that can be wholly contained with a tetrahedron of side.

E2453.741.S752.() Determine all rational numbers r for which 1, cos 2πr and sin 2πr are linearly dependent over the rationals.

E2471.74.S755.() Let ma, wa and ha denote the median, angle bisector and altitude to side a of ABC respectively. Show athat (b + c)2 m b2 + c2 m ≤ a , ≤ a . 4bc wa 2bc ha When does equality hold?

E2475.74.S756.() Under what conditions can the four tritangent circles of a triangle be rearranged so as to be mutually tangent?

E2477.74.S756.() A straight line L meets the sides BC,CA and AB of ABC with orthocenter H at X, Y , Z respectively. DE is a diameter of the circle ABC Through X, Y , Z lines BC, CA and AB are drawn parallel to AE, BE CE to form a triangle ABC oppositively similar to ABC.IfD, E H are the images of D, E, H. (a) The lines AA, BB CC, DD, HH concur, so that ABCD and ABCD (ABCH and ABCH) are oppositely similar perspective cyclic (orthocentric) quadrangles; (b) The lines DH and HD meet at the invariant point of the similarlity, and DHDH is a cyclic quadrangle; (c) The axis L is perpendicular to DD and bisects EE.

E2498.74?.S765.(R.E.Smith) Given triangle ABC, ﬁnd the locus of all points R (not necessarily in the plane of ABC) with the property that the three triangles RAB, RBC and RCA havethesamearea.

E2501.74.S759.() Let ABC be a triangle with C ≥ B ≥ A. Show that BIO is a right triangle if and only if a : b : c =3:4:5.

√ 1 h + m + t 3 ≤ a b c ≤ ; 4 a + b + c 2 3 t + m + m ≤ a b c ≤ 1. 8 a + b + c

E2505.74?.S761.(Garfunkel) Extend the medians of a triangle to meet the circumcircles again, and let these chords be Ma, Mb, Mc respectively. SHow that See also E2959.

4 M + M + M ≥ (m + m + m ); a b c 3 a b c 2√ M + M + M ≥ 3(a + b + c). a b c 3 When does equality occur ?

E2512.751.S762.(E.A.Herman) Let T1 and T2 be two triangles with circumcircles C1 and C2 respectively. Show that if T1 meets T2 then some vertex of T1 lies in (or on) C2 or vice versa. Generalize.

E2513.751.S762.(N.Felsinger) Let P be a simple (non-self-intersecting) planar polygon. If A is a point in the plane, and if E is an edge of P ,thenE is viewable from A if for every point x of E, the line segment joining A to x contains no point of P other than x. (a) Let A and P be arbitrary. Must some edge of P be viewable from A ?Examinethe cases of A exterior to P and interior to P separately. (b) Find suﬃcient conditions of A in order that some edge of P is viewable from A.

|M| =(p − q)2|P | +4pq|K|, where |M| denotes the area of M etc.

E2531.754.S766.(V.F.Ivanoﬀ) Given points A, B, C, D, E, F in the plane, let ABC denote the directed area of triangle ABC,provethat

AEF · DBC + BEF · DCA + CEF · DAB = DEF · ABC.

E2576.752.S775.(R.L.Helmbold) What is the area of the orthogonal projection of the x 2 y 2 z 2 ellipsoid ( a ) +(b ) +(c ) = 1 onto a plane perpendicular to the unit vector Gn =(n1,n2,n3)?

E2625.7610.S782.(H.Demir) Let Ai, i =0, 1, 2, 3(mod4),befourpointsonacircleΓ. Let ti be the tangent to Γ at Ai and let pi and qi be the lines parallel to ti pasing through the points Ai−1 and Ai+1 respectively. If Bi = ti ∩ ti+1,andCi = pi ∩ qi+1, show that the four lines BiCi have a common point.

E2657.775.S788.(G.Tsintsifas) Let A = A0A1 ···An and B = B0B1 ···Bn be regular n−simplices in Rn. Assume that the ith vertex of B lies on the ith face of A,0≤ i ≤ n.What is the minimal value of their similarity ratio ?

E2669.777.S7810.(I.J.Schoenberg) Let a>b>0. For a given r,0

··· ··· − E2674.778.(G.Tsintsifas) Let S = A0A1 An and S = A0A1 An be regular n simplices such that Ai lies on the opposite face of Ai. Is it true that the centroids of S and S coincide ?

E2680.779.S792.(J.W.Grossman) Let ABCD be a convex quadrilateral in the hyperbolic plane. Assume that AD = BC and that

A + B = C + D.

Does AB = CD follow from the above hypotheses ? (It does in the euclidean plane). YIU : Problems in Elementary Geometry 144

E2682.779.S793.(D.Hensley) Let E be an ellipse in the√ plane whose interior area A ≥ 1. Prove that the number n of integer points of E satisﬁes n<6 3 A.

E2687.7710.S799.(R.Evans) Does there exist a triangle with rational sides whose base equals its altitude ? Answer. No.

E2701.783.S795.(R.Stanley) Find the volume of the convex polytope determined by xi ≥ 0, 1 ≤ i ≤ n and xi + xi+1 ≤ 1, 1 ≤ i ≤ n − 1.

E2715.785.S798,804.(Garfunkel) Let G be the centroid of ABC. Prove or disprove 3 sin GAB +sinGBC +sinGCA ≤ . 2 The inequality is true.

E2736.788.S822.(E.Ehrhart) Let be a closed triangle and P0,A0,P1,A1,... an inﬁnite sequence of points in a plane. Assume that Pi = Pi+1, Ai = Ai+1,eachAi is a vertex of and YIU : Problems in Elementary Geometry 145

the midpoint of the segment [Pi,Pi+1], and that [Pi,Pi+1] ∩= {Ai}.ProvethatPn = P0 for some positive n.

S2.791.S802.(Coxeter) In the hyperbolic plane, the locus of a point at constant distance δ from a ﬁxed line (on the side of the line) is one branch of an ‘equidistant’ curve (or hypercycle). In Poincar´e’s half-plane model, this curve can be represented by a ray making a certain angle with the bounding line of the half-plane. Show that this angle is equal to Π(δ), the angle of parallelism for the distance δ.

E2751.791.S814.(P.Monsky) Let X be a conic section. Through what points in space do there pass three mutually perpendicular lines, all meeting X ?

E2757.792.(H.D.Ruderman) Let a, b, c be three lines in R3.FindpointsA, B, C on a, b, c respectively such that AB + BC + CA is a minimum.

S12.795.S807.(Klamkin) If a, a1; b, b1; c, c1 denote the lengths of three pairs of opposite sides of an arbitrary tetrahedron, prove that a + a1,b+ b1,c+ c1 satisfy the triangle inequality.

S16.797.(I.J.Schoenberg) Characterize the closed sets S of the complex plane such that d(z + w) ≤ d(z)+d(w) for all complex numbers z and w,whered(z) denotes the euclidean distance from z to S.

∗ H.Flanders

E2793.798.S819.(E.D.Camier) P and Q are two points isogonally conjugate with respect to a triangle ABC of which the circumcenter, orthocenter, and nine-point center are O, H,and N respectively. If OR = OP + OQ,andU is the point symmetric to R with respect to N,show that the isogonal conjugate of U in the triangle ABC is the intersection V of the lines P1Q and PQ1 where P1 and Q1 are the inverses of P and Q in the circle ABC. (Assume that neither of P , Q is on the circle ABC).

E2802.799.S811.(M.Slater) Given a triangle ABC (in the euclidean plane), construct similar isosceles triangles ABC, ACB outwards on the respective bases AB and AC,andBCA inwards on the base BC (or ABC and ACB inwards and BCA outwards). Show that ABAC (respectively ABAC) is a parallelogram.

E2831.805.S819.(M.Cavachi) Prove that a convex hexagon with no side longer than 1 unit must have at least one main diagonal not longer than 2 units.

E2836.807.S825.(J.E.Valentine) Show that an absolute geometry (no parallel postulate) is euclidean (or riemannian) if some triangle has the property that a median and the segment YIU : Problems in Elementary Geometry 147 joining the midpoints of the other two sides bisect each other.

(124) = a1, (235) = a2, (341) = a3, (452) = a4, (513) = a5.

Determine the area of the pentagon A1A2A3A4A5. The analogous problem, with (123), (234), (345), (451) and (512) being given, was solved by Gauss in 1823. See Crux 3 (1977), p.240.

E2843.807.(P.Ungar) A set of√ nonoverlapping rectangles, each having its longer side equal to 1, is inside a circle of diameter 2. Show that the sum of their area is ≤ 1.

E2848.808.S825.(J.Fickett) Prove that the regular tetrahedron has minimum diameter among all tetrahedra that circumscribe a given sphere. (The diameter is the length of a longest edge).

E2866.811.S826.(J.Dou) Let AKL, AMN be equilateral triangles. Prove that the equilateral triangles LMX, NKY are concentric (if Y is on the properly chosen side of NK).

E2872.813.S82?*****

≥ ≤ ◦ E2874.813.S828.(N.Kimura and T.Sekiguchi) Let n 3, 0

E2894.816.(T.Ihringer) Let n be ﬁxed. In how many ways can a square be dissected int (a) n congruent rectangles, (b) n rectangles of equal area ?

2917.8110.S846.(F.W.Luttman) Let P0 be a convex polygon of n sides and let 0

(1) Pk+1 is a linear contraction of Pk by the factor f. (2) Two adjacent sides of Pk+1 lie on Pk. (Necessarily Pk and Pk+1 share a single vertex). (3) The vertex which Pk shares with Pk+1 lies next clockwise from the vertex it shares with Pk−1. There is precisely one point lying inside all PK ’s. Construct it. (See Coxeter, Introduction to Geometry, p.164).

|A1M1 − A2M2|≤|A1T1 − A2T2|.

E2930.822.S842.(Monthy Problem Editors) Find the largest square that can be inscribed in some triangle of area 1. See also E3114.

E2958.827.S854.(Klamkin) Let x, y, z be positive, and let A, B, C be angles of a triangle. Prove that x2 + y2 + z2 ≥ 2yz sin(A − 30◦)+2zxsin(B − 30◦)+2xy sin(C − 30◦).

E2967.828.(J.Dou) Divide a circle into four equiareal parts with (i) arcs (ii) segments of minimal total length.

E2968.829.S855.(G.Tsintsifas) The points A1,A2,A3 lie on the sides A2A3, A3A1, A1A2 of an acute angle triangle A1A2A3 respectively. Show that ≥ 2 ai cos Ai ai cos Ai where a1,a2,a3 are the sides of the triangle A1A2A3 and a1,a2,a3 are the sides of the triangle A1A2A3.

E2983.831.S897.(E,Ehrhart) Let ABC be an equilateral triangle of perimeter 3a.Calcu- late the area of the convex region consisting of all points P such that PA+ PB+ PC ≤ 2a. YIU : Problems in Elementary Geometry 151

E2992.834.(J.Dou) Find the shape of a contour of length L that encloses the largest possible area and is constrained to pass through three given points.

E3009.837.S8610.(C.Jantzen) Points X, Y, Z are chosen on the sides of ABC such that

AX BY CZ = = = k XB YC ZA and a triangle PQR is formed using CX,AY,BZ as sides. The operation is repeated on PQR, that is the points X.Y ,Z are chosen on the sides of PQR such that

PX QY RZ = = = k XQ Y R ZP YIU : Problems in Elementary Geometry 152 and a triangle LMN is formed using RX,PY,QZ as sides. Show that LMN is similar to ABC and ﬁnd the ratio of similarity.

E3038.842.S869.(T.Sekiguchi)√ Prove the inequalities. ≤ 3 3 (a) sin A +sinB +sinC√ 2 . ≤ 3 3 (b) sin A sin B sin C 8 .

E3044.845.S871.(J.Dou) Construct ABC given r, AI, AH.

E3054.848.S8610.(V.D.Mascioni) Prove the inequalities. YIU : Problems in Elementary Geometry 153

≥ 2π 2 (a) abcABC ( 3 ) r . √ − − − ≥ 4π 3 3 (b) abc(π A)(π B)(π C) ( 9 ) s .

E3071.851.S882.(K.Satyanarayana) In ABC, A < B < C.ProvethatI lies inside OBH.

E3080.853.(L.C.Larson) Can the following equations be satisﬁed with integers?

(x +1)2 + a2 =(x +2)2 + b2 =(x +3)2 + c2 =(x +4)2 + d2.

E3091.855.S875.(C.P.Popescu) Equilateral triangle ABC is inscribed in XY Z,withA between Y and Z, B between Z and X,andC between X and Y . Show that

XA + YB+ ZC < XY + YZ+ ZX.

Is equality possible? YIU : Problems in Elementary Geometry 154

E3114.859.S881.(M.J.Pelling) Find the largest cube that can be inscribed in some tetrahedron of volume 1. See also E2930.

E3134.862.S885.(J.Dou) Construct ABC given ma,wa,A.

E3146.864.S888.(J.J.Wahl) Prove or disprove: √ √ √ 2s( s − a + s − b + s − c) ≤ 3( bc(s − a)+ ca(s − b)+ ab(s − c)). The statement is correct.

E3150.865.S887,9010.(G.A.Tsintsifas) For arbitrary positive real numbers p, q, r, p q r √ a2 + b2 + c2 ≥ 2 3. q + r r + p p + q

E3154.866.S887.(G.A.Tsintsifas) Let A1,B1,C1 be points on the sides a, b, c of ABC respectively, and a1,b1,c1 the sides of A1B1C1.Provethat 2 2 2 2 a b1c1 + b c1a1 + c a1b1 ≥ 4 . YIU : Problems in Elementary Geometry 155

E3157.866.S886,8910.(L.I.Nicolaescu) How many sets of four distinct points forming the vertices of a trapezoid are there if the points are chosen from the vertices of a regular n−gon, n ≥ ?

E3164.867.S887.(H.Demir) Let s, t be the lengths of the tangent line segments to an ellipse s from an exterior point. Find the extreme values of the ratio t .

E3167.868.S9010.(E.B.Cossi) Let Pi be any one the ﬁve regular polyhedra inscribed in a unit sphere. For each polyhedron Pi, determine the smallest and the largest number of vertices of Pi which can be seen from a point on a concentric sphere of radius R>1.

E3172.869.S8810,8910.(J.Dou) Let A (respectively B, C be the foot of the altitude from vertex A (respectively B, C), in a triangle ABC.LetH be its orthocenter, adn M be an arbitrary point of the plane. Prove that the conics MABAB, MBCBC, MCACA, MHCAB, MHABC and MHBCA have a common point other thatn M.

E3180.8610.S888.(Klamkin) Prove that

A B C A B C cos +cos cos 1+sin +sin sin . 2 2 ≥ 2 2 2

E3185.871.S898.(R.E.Spaulding) Let P be a point in the interior of an equilateral triangle, and let S be the sum of the perpendicular distances to the three sides of the triangle from P . In euclidean geometry, the sum S always equals the altitude of the triangle. In Lobachevskian YIU : Problems in Elementary Geometry 156 geometry, prove that S is less than any altitude. In addition, ﬁnd the position of P which would give a minimum value for S.

E3195.873.S896.(L.Kuipers) Given ABC, consider those inscribed ellipses touching AB in C1, BC in A1 and CA in B1 with AB1.BA1 = B1C.A1C. Describe the locus of the centers of such ellipses.

(AC)3 = AQ · BC · CS.

E3208.873.S891.(I.D.Berg, R.L.Bishop, and H.G.Diamond) Suppose that the euclidean plane, line segments of lengths a, b, c, d emanate from a given point P in clockwise order, where a, b, c, d are given positive numbers with

a2 + c2 = b2 + d2.

(i) Show that the four segments can be so placed that the endpoints determine a rectangle containing P , and show that this rectangle may have any speciﬁed area between 0 and some maximum value M(a, b, c, d). (ii) Determine M(a, b, c, d). See also MG1147.

E3209.873.S893.(B.Reznick Family of circles. Let C0,C1,C2,... be the sequence of circles in the Cartesian plane deﬁned as follows: 2 2 (i) C0 is the circle x + y =1; (ii) for n =0, 1, 2,...,thecircleCn+1 lies in the upper half - plane and is tangent to Cn as well as to both branches of the hyperbola x2 − y2 =1. Let rn be the radius of Cn. Show that rn is an integer and give a formula for rn.

PiQi − signed ration λi so that if si meets pi 1 in Qi,thenλi = QiPi+1 . Prove that the lines s1,s2,s3 are concurrent if and only if λ1λ2λ3 − (λ1 + λ2 + λ3)=2.

E3236.879.S906.(N.D.Elkies) For a plane triangle call two circles within the triangle companion incircles if they are the incircles of the two triangles formed by dividing the given triangle by passing a line through one of the vertices and some point on the opposite side. (a) Show that any chain of circles C1,C2,... such that Ci and Ci+1 are companion incircles for every i consists of at most six distinct circles. (b) Give a ruler and compass construction for the unique chain which has three distinct circles. (c) For such a chain of three circles show that the three subdividing lines are concurrent. See AMM 10780.

E3239.8710.S906.(Klamkin) Show that if A is any three dimensional vector and B, C are unit vectors, then [(A + B) × (A + C)] × (B × C) · (B + C)=0. Interpret the result as a property of spherical triangles.

E3256.883.S901.(J.Isbell) (a) Let T be the set of triangles in the plane whose vertices have integral coordinates and whose sides have integral lengths. Certain isosceles triangles in T can be constructed by ﬁtting together two congruent right triangles in T , e.g. the isosceles triangle with vertices (−12, −9), (12, 9), (−12, 16) arises in this way. Are there any other isosceles triangles in T ? (b) Consider the set V of triangles in 3-sapce whose vertices have integral coordinates. Does V contain any equilateral triangles with integral side-length ?

E3257.883.S911.(I.A.Sakmar) Let P, Q, R be the new vertices of equilateral triangles constructed outwardly on the edges of a given triangle ABC. YIU : Problems in Elementary Geometry 158

(a) Show that any triangle PQR which can be obtained in this way arises from a unique triangle ABC, and give a construction for recovering triangle ABC from triangle PQR. (b) Show tht not every triangle PQR can be so obtained.

E3270.886.S899.(B.Lindstr¨√ om) Determine those positive rational numbers m for which arctan m is a rational multiple of π.

E3279.887.S901.(S.Rabinowitz) For some n ﬁnd a simplex in En with integer edges and volume 1. See also MG871.1261.S882

E3282.888.S9010.(D.M.Milosevic) Prove the inequality

R w2 + w2 + w2 ≤ s2 − r( − r), a b c 2 with equality if and only if the triangle is equilateral.

E3283.888.S909.(O.Frink) Every simple closed polygon in the plane has three centroids, namely the centroid of its vertex set, the centroid of its boundary, and the centroid of its interior. In general, all three are distinct. (a) In the case of a triangle show that these centroids coincide if and only if the triangle is equilateral. (b) Which of the three centroids are aﬃne invariant ?

E3299.8810.S911.(J.Yamout) Suppose ABCD is a plane quadrilateral with no two sides parallel. Let AB and CD intersects at E and AD, BC intersect at F .IfM,N,P are the midpoints of AC, BD, EF respectively, and AE = a · AB, AF = b · AD,wherea and b are nonzero real numbers, prove that MP = ab · MN.

E3307.892.S918.(P.Andrews, Klamkin, and E.T.H.Wang) The celebrated Morley triangle of a given triangle ABC is the equilateral triangle whose vertices are the intersections of adjacent pairs of internal angles trisectors of ABC. Determine the maximum values of (i) sM /s, (ii) RM /R, (iii) rM /r, (iv) M /.

E3314.893.S909.(A.Bege) Let P be a point inside acute triangle ABC.Putα1 = PAC,β1 = PBA,γ1 = PCB.Provethat

3 1 1 cot α1 +cotβ1 +cotγ1 > 2 2 3 4 (cot B +cotB +cotC) 2 .

E3337.897.S9010.(Klamkin) Suppose the two longest edges of tetrahedron are a pair of opposite edges. Prove that the three edges incident to some vertex of the tetrahedron are congruent to the sides of an acute triangle.

E3369.902.S918.(J.Fukuta) Suppose we are given a piece of paper in the shape of an equilateral triangle. Suppose P is a point in the intersection of the three open circular discs with diameters AB, BC, CA. If we fold the three corners of the paper in such a way that the vertices coincide with P , we get a hexagon three sides of which are the creases formed and the other three sides of which are portions of the sides of ABC. Prove that the area and perimeter of this hexagon are both maximized when P is the centroid of ABC.

E3375.903.(R.J.Chapman) Express ∞ 1 tan arctan 2 n=1 n in closed form. YIU : Problems in Elementary Geometry 160

ABC ≥ 16A2B2C2, and determine when equality holds.

E3397.907.S921.(J.Chen and C.H.Lo) The perimeter of ABC is divided into three equal parts by three points P, Q, R. Show that 2 PQR> ABC, 9

2 and that the constant 9 is best possible.

E3407.909.S924.(C.Kimberling) Suppose ABC is a given triangle. Prove the existence of a triangle that is in perspective with every antipedal triangle of ABC. Answer: The circumcevian triangle of O. Trivially, ABC is also an answer.

π Answer. 2 . Beautiful solution by R.M.Robinson.

E3417.911.S927.(R.S.Luthar) Suppose ABC is a triangle with AB = AC,andletD, E, F, G be points on the line through B and C deﬁned as follows: D is the midpoint of BNC, AE is the bisector of BAC, F is the foot of the perpeandicualr from A to BC,andAG is perpendicular to AE (i.e. AG bisects one of the exterior angles at A). Prove that AB · AC = DF · EG. Solution. Assume b>c. ac ac 2abc 2abc EG = EB + BG = + = = b + c b − c b2 − c2 CF2 − BF2 2abc 2abc bc = = = . (CF + BF)(CF − BF) 2a · DF DF

E3422.912.S927.(H.Demir and C.Tezer) Suppose F and F are points situated symmetrically with respect to the center of a given circle, and suppose S is a point on the circle not on the line FF.LetP and P be the second points of intersection of SF and SF respectively with the circle. If the tangents to the circle at P and P intersect at T , prove that the perpendicualr bisector of FF passes through the midpoint of the line segment ST.

E3434.914.(J.P.Hoyt) Prove or disprove that there are inﬁnitely many triples of positive integers (a, b, c) with no common factor such that the triangle ABC with sides a, b, c has the following property: the median from A, the angle bisector from B and the altitude from C are concurrent. Examples of such triples are (12, 13, 15), (35, 277, 308), (26598, 26447, 3193). See CMJ455.913.S923.

E3438.914.S929.(H.G¨ulicher) Let P1P2P3 have the longest side P1P2. For each of the six permutations of 1,2,3, let Pij be the point on the ray PiPj such that PkPiPij = PiPjPk. Let pij be the length of PkPij and let pi be the length of PjPk.Provethat 2 2 2 p12 p21 (i) p1 + p2 = p3 if and only if p13 + p23 =1; 3 3 3 p31 p32 (ii) p1 + p2 = p3 if and only if p13 + p23 =1.

E3468.919.(C.Cooper, R.E.Kennedy, and S.Rabinowitz) Suppose m and n are positive integers such that all primes factors of n are larger than m. (a) Prove that n kπ φ(n) − µ(n) 2m ∗ sin2m( )= , n 4m m k=1 where * denotes summation over integers relatively prime to n. (b) Find a similar formula for cosines.

American Mathematical Monthly Advanced Problems, 1976 – 1995.

5k2 − 4k +4 k(k2 − 4k + 20) k +2 , , k2 − 4 2(k2 − 4) 2 are positive and form the sides of a triangle with area k.Thecase0

AMM5790.71?.S823.(D.E.Daykin) Find all nontrivial maps f : R2 → R2 such that whenever a, b, c are collinear, then f(a),f(b),f(c) are collinear.

AMM5986.74?.S764.(D.E.Daykin) Let E be the real euclidean plane and 0 <α<1. What can be said about maps f : E → E which send each triangle T into a triangle fT with area fT ≤ α area T ?

AMM6053.75?.S77?. (R.Finkelstein)

AMM6062.75?.S777,812,827.(B.H.Voorhees) Consider an inﬁnite sequence of regular n−gons such that each(n +1)− gon is contained within the preceding n−gon and is of maximal area consistent with this constraint. Take the ﬁrst element of this sequence as an equilateral YIU : Problems in Elementary Geometry 164 triangle having unit area. Is the limit of this sequence a point or a circle ? If it is a circle, determine its area.

AMM6179.779.S8010.(E.Ehrhart) Find all cubes in a cubic lattice whose vertices are lattice points.

AMM6367.819.S838.(A.Ehrenfeucht and J,Mycielski) Let A be a ﬁnite collection of YIU : Problems in Elementary Geometry 165 distinct but possibly overlapping regular n−gons of the same size on the plane such that every vertex of every n−gon of A is a vertex of exactly two n−gons of A. (a) Construct a collection A of 2nn−gons such that, even if the n−gons are rigid, A is ﬂexible. (b) For which n is rigid A possible ?

AMM6377.822.(K.R.Kellum) Suppose G is a subset of the euclidean plane such that G meets each vertical line in exactly two points and G meets each nonvertical line in a dense set of points. Must G have a subset H such that H meets each vertical line in one point and each nonvertical line in a dense set of points ?

AMM6381.823.(W.W.Meyer) M.D.Fox (AMM 87 (1980) 708 – 715) defines a Steiner chain as a sequence of circles each touching its two neighbours and two given boundary circles CO and CI . Enlarging on this, with the hypothesis that CO suurounds CI , we define a linear chain as a polygon circumscribed by Co and circumscribing CI . Linear or circular, a chain is n said to have period m if it closes on itself, the first link and the (n+1)th link coinciding, after m cycles arounds C0. Proe that a linear chain of rational period p and a circular chain of rational period q coexists if and only if π π π π p>2,q>2, cos cos( − ) ≤ cos 2p 4 2q 4 and then the boundary circles are uniquely determined as to relative size and eccentricity.

AMM6385.824,S839.(M.D.Meyershon) Prove or disprove: every simple closed curve in euclidean space contains the vertices of a rectangle. (It is known to be true in the euclidean plane).

AMM6477.84?.S86?.8910.() Let r be the radius of the√ incircle of an arbitrary triangle ≤ 5−1 lying in the closed unit square. Prove or disprove that r 4 . Original published solution is incorrect.

AMM6418.831.S848.(G.Benke) Prove that

2N−1 πn2 sin 2N πn = N. n=1 sin 2N YIU : Problems in Elementary Geometry 166

AMM6477.849.S865,8910.(L.Funar) Let r be the radius of the√ incircle of an arbitrary ≤ 5−1 triangle lying in the closed unit square. Prove or disprove that r 4 .

AMM6478.849.S909.(L.Funar) Let r be the radius of the incircle of an arbitrary triangle lying in a closed ﬁgure F of width w,andletR be the radius of the incircle of F .Arethe following inequalities valid ? 1 ≤ sup r ≤ 1 (a) 4 w 2 ; 1 ≤ sup r ≤ (ii) 2 R 1.

AMM6557.879.S906.(C.Kimberling) Let C be the circumcircle of ABC.LetA be the point, other than A,wheretheA-median of ABC meets C.LetA,bethepoint,otherthan A,wheretheA-altitude of ABC meets C. Similarly deﬁne B,C nd B,C.LetDEF be the tangential triangle of ABC (D is the point where the line tangent to C meets the line tangent to C at C). Prove that the lines DA,EB,FC and the lines DA,EB,FC concur in points that lie on the Euler line of ABC.

AMM6560.8710.S898.(A.J.Krishna,M.M.Rao, and G.S.Rao) If x and y are odd positive integers, evaluate ∞ 1 nπ nπ 2 tan tan . n=1 n x y

AMM6605.896.(E.Ehrhart) If k is a poisitive integer, Schinzel, Enseignement Math. (2) 4 (1958) 71 – 72, proved that the circle 1 2 5k 2 X − + Y 2 = 3 3 passes through exactly 2k + 1 lattices points; clearly, the two coordinates of any one of these 2k + 1 lattice points are of like parity. Thus, by making the substitution X = x + y, Y = x − y, we see that the smaller circle √ 1 2 1 2 5k 2 2 x − + y − = (∗) 6 6 6 YIU : Problems in Elementary Geometry 167 also passes through exactly 2k + 1 lattice points. (i) Show that when k = 1 no circle smaller than (*) passes through exactly 3 lattice points. (ii) Show that when k = 2 no circle smaller than (*) passes through exactly 5 lattice points. (iii) Show that when 2k + 1 is composite, there is a circle smaller than (*) which passes through exactly 2k + 1 lattice points.

AMM6628.904.S918.(R.A.Melter) Call a triangle a Heron triangle if it has integer sides and integer area. Fermat showed that there does not exist a Heron right triangle whose area is a perfect square. However, the triangle with sides 9, 10, 17 has area 36. Prove that there are inﬁnitely many Heron triangles whose sides have no common factor and whose area is a perfect square. Solution. (C.R.Maderer) For each positive integer k, deﬁne

a(k):=20k4 +4k2 +1, b(k):=8k6 − 4k4 − 2k2 +1, c(k):=8k6 +8k4 +10k2.

Here, a(k),b(k)

[(2k)(2k2 − 1)(2k2 +1)]2.

1−t1t2 are rational. Note that t3 = t1+t2 . The area of the triangle is a rational square if and only if

2 u = t1t2(t1 + t2)(1 − t1t2).

1 The conclusion follows from the observation that elliptic curve for t2 = 4 has positive rank. YIU : Problems in Elementary Geometry 168

(Editor’s comment): N.Elkies, On A4 + B4 + C4 = D4, Math. Comp/ 51 (1988) 825 – 835, has proved that this equation has inﬁnitely many solutions with gcd(A, B, C, D) = 1. Since fourth powers are congruent to 1 or 0 modulo 16, in such a quadruple, exactly one of A, B, C is odd. If we choose a = B4 + C4, b = A4 + C4,andc = A4 + B4,thengcd(a, b, c) = 1, and a, b, c form a triangle with are (ABCD)2. YIU : Problems in Elementary Geometry 169

American Mathematical Monthly 1992-1999

AMM 10202.923.S939.(J.B.Romero M´arquez) Let A,B,C be the feet of the altitudes of ABC and let X, Y, Z be the centers of the circumscribing rectangles of ABC with edges BC, CA, AB respectively. Prove that XY Z is a dilation of ABC.

AMM 10244.927.S945.(K.Bromberg and Stan Wagon) A classical construction of Miquel starts with an n−vertex polygon and a point P in the plane (not a vertex of the n−gon), and forms another n−gon as follows: (1) draw the perpendiculars from P to the (extended) sides of the polygon; (2) connect the feet to obtain another n−gon. These steps are then repeated n times (provided that none of the polygons has P as a vertex). The resulting polygon, denote M(P ) is similar to the initial n−gon. (a) Given a triangle, construct the point P for which M(P ) is largest. (b) Given a quadrilateral, is there a euclidean construction of the point P for which M(P ) is largest ?

AMM 10249.928.S948.(O.Yumlu) Suppose that the inradius of an isosceles triangle and the ratio of the distances from its incenter to its vertices are given. Give a euclidean construction of the triangle. Solution. Let ABC be the desired isosceles triangle with AB = AC, incenter I and inradius r. Construct an isosceles right triangle IXY with IX = XY = r and a right angle at X.LetX ,X be points on the line IX such that IX : IX = k = the given√ ratio IA : IB. The line through X parallel to X Y meets the√ IY at a point Y such that IY = 2kr.Onthe line XY , mark a point P with XP = IY = 2kr.LetC, C be the circles with center I passing through X and P respectively. Note that the tangents from any point on C to C has square length 2k2r2. Mark a point Q on C with XQ = r, and extend XQ to meet C at K. Finally, extend XI to a point A such that IA = XK. The triangle bounded by the tangents to C from A and at X is the desired isosceles triangle. To justisfy the construction, let α and β be the angles at A and B respectively. Since IA(IA − r)=KX(KX − QX) = the square length of the tangents from K to C =2k2r2,we have

r · r − 2 2 α α r =2k r , sin 2 sin 2 YIU : Problems in Elementary Geometry 170

α α 1 − sin =2k2 sin2 . 2 2 α π Since 2 + β = 2 , α β sin =cosβ =1− 2sin2 . 2 2 β · α · From (1) and (2) it follows that sin 2 = k sin 2 ,andIA = k IB.

AMM 10251.929.S955. (J.G.Mauldon) Let C denote the unit cube, and let P be the set of all pairs [a, b]witha and b mutually perpendicular line segments contained in C. (a) Evaluate sup{min{|a|, |b|} :[a, b] ∈ P}. (b) Deduce the area of the largest square, and the volume of the largest regular octahedron, that ﬁt into C.

AMM 10254.929.S954. (E.Erhart) The curve traced out by a ﬁxed point of a closed convex curve as that curve rolls without slipping along a second curve wil be a called a “roulette”. Let S be the area of one arch of a roulette traced out by an ellipse of area s rolling on a straight line. Prove or disprove that S ≥ 3s, with equality only if the ellipse is a circle.

AMM 10256.929.S945. (Klamkin) Let Ai, Ai, i =1, 2, 3, 4, be the vertices of a rectangular P P parallelepiped ,withAi diametrically opposite to Ai.Let P be any interior point of .Prove that ≤ · · · · S 2(PA1 PA1 + PA2 PA2 + PA3 PA3 + PA4 PA4) where S denotes the surface area of P.

AMM 10269.9210.S951.(D.M.Bloom) Prove that there is constant K<1 with the following property. Let G be a regular (2m +1)−gon inscribed in the unit circle, and let any point P ∈ G be given, then there are distinct vertices V0 and V1 of G such that K |d(P, V ) − d(P, V )|≤ . 0 1 m

AMM 10275.931.S951.(Klamkin) Let A be a regular n−gon with edge length 2. Denote the consecutive vertices by A0,A1,...,An−1 and introduce An as a synonym for A0.LetB be a regular n−gon inscribed in A with vertices B0,B1,...,Bn−1 where Bi lies on AiAi+1 and |AiBi| = λ<1for0≤ i

AMM 10282.932.S955. (Erd¨os) Let A, B, C be the vertices of a triangle inscribed in a unit circle, and let P be a point in the interior of the triangle ABC. Show that 32 PA· PB · PC < . 27

AMM 10293.933.S953. (M.Rosenfeld) Suppose four distinct lines through the origin in R3 have the peroperty that the six acaute angles between pairs of these lines are all equal. Prove that this conﬁguration of four lines is isometric either to the diagonals of a cube or to a conﬁguration of four of the the six diagonals of a regular icosahedron.

AMM 10303.934.S9410.(D.E.Gurarie) Let a1,a2,...,an be positive real numbers. (a) Find necessary and suﬃcient conditions on these numbers for there to exist a convex n−gon which admits an inscribed circle and whose sides, in cyclic order, are a1,...,an. (b) Find the radius of the inscribed circle.

AMM 10308.935.S965.(R.Connelly, J.H.Hubbard and W.Whitely) Suppose that p1, p2, p3, q1, q2, q3 are six points in the plane and that the distance between pi and qj, i, j =1, 2, 3, is i + j. Show that the six points are collinear.

AMM 10317.936.S9610. (J.B.R.M´arquez) Let ABC be inscribed in a circle C and let A,B,C be the midpoints of the arcs BC, CA, AB respectively. (a) Prove that the incenter of ABC is the orthocenter of ABC. (b) Prove that the pedal triangle of ABC is homothetic to ABC.

AMM 10322.937.S967. (Jiang Huanxin) Let ABCD and AEF G be squares with the π common vertex A and diﬀerent edge lengths. Let θ = EAD,0<θ< 2 . Suppose that EF and CD intersect at the point P . For which value of θ will AP be perpendicular to CF ?

AMM .10344.939.(E.Ehrhart) Let S be a regular tetrahedron, and let P ∈ S. Deﬁne DV (P ) to be the sum of the distances from P to the vertices of S,andDE(P )tobethesumof the distances from P to the edges of S. Find the maximum and minimum values of DE(P ) . DV (P )

AMM 10348.9310.S971. (Jiang Huanxin) Let D, E, F be distinct points on the sides BC, CA, AB respectively of ABC.Letα = BDF, β = FDA, γ = ADE,andδ = EDC. α δ If AD, BE and CF are concurrent and β = γ = m =1,provethatα = δ and β = γ.

AMM 10358.941.S977. (Jiang Huanxin) In triangle ABC, ﬁnd all points P such that the triangle DEF (with D = AP ∩ BC, E = BP ∩ CA, F = CP ∩ AB) is equilateral. YIU : Problems in Elementary Geometry 172

AMM 10368.943.S978. (E.Alkan) For each point O on diameter AB of a caircle, perform the following construction. Let the perpendicular to AB at O meet the circle at point P .Inscribe circles in the ﬁgures bounded by the circle and the lines AB, OP. Let (the) R and S be the points at which the two incircles to the curvilinear triangles AOP and BOP are tangent to the diameter AB. Show that RPS is independent of the position of O.

AMM 10371.943.S972. (E.Y.Stoyanov) Let B and C be points on the sides AB and AC respectively of a given triangle ABC,andletP be a point on the segment BC. Determine the maximum value of min{[BPB], [CPC]} [ABC] where [F ] denotes the area of F .

AMM 10374.943.(D.L.Book) Given an integer N, characterize the smallest square in the plane containing N lattice points. See editorial notes on narrative on this problem.

AMM 10378.944.S978. (B.Poonen) Given that point D is in the interior of ABC and that there are real numbers a, b, c, d such that AB = ab, AD = ad, BC = bc, BD = bd and | | | | π CD = cd.Provethat ABD + ACD = 3 .

AMM 10386.945.S996.(J.Tabov) Let a tetrahedron with vertices A1,A2,A3,A4 have altitudes that meet in a point H. For any point P ,letP1,P2,P3,P4 be the feet of the perpendiculars from P to the opposite faces of A1,A2,A3,A4 respectively. Prove that there exist constants a1,a2,a3,a4 such that one has

PH = a1PP1 + a2PP2 + a3PP3 + a4PP4 for all points P .

AMM 10405.948.(H.G¨licher) Let A1A2A3A4A5A6 be a hexagon circumscribed about a conic, and form the intersections Pi of AiAi+2 with Ai+1Ai+3, i =1, 2,...,6. Show that the Pi are the vertices of a hexagon inscribed in a conic.

AMM 10413.949.S988. (M.Mocanu) Four disjoint (except for boundary points) equilateral triangles of sides a, b, c,andd are√ enclosed in a regular hexagon of unit side. (a) Prove that 3a + b + c + d ≤√4 3. (b) When is 3a + b + c + d =4 3? √ (c) Prove or disprove that a + b + c + d ≤ 2 3. YIU : Problems in Elementary Geometry 173

AMM 10415.949.S979. (Kitchen) Let A be a triangle whose centroid is at the origin. Choose k ∈ R, and dilate one of the Napoleon triangles of A by a factor of −k and the other by k A afactorof 1−k .Provethat is simultaneously perspective with both dilated triangles.

AMM 10418.9410.S983. (R.Satnoianu) Given that acute triangle ABC,letha,hb and hc denote the altitudes and s the semiperimeter. Show that √ 3max(ha,hb,hc) ≥ s.

AMM 10440.953. (M.Cavachi) Show that the euclidean plane cannot be covered with circular disks having mutually disjoint interiors.

AMM 10453.956.S986. (Klamkin) Prove that the following two properties of the altitudes of an n−simplex are equivalent: (a) the altitudes are concurrent; (b) the feet of the altitudes are the orthocenters of their respective faces.

AMM 10455.955.S988. (Z.Franco) It is easily seen that a parabola can intersect a circle in at most 4 points. (a) Show that there is a number R such that a regular polygon (of any number of sides) can intersect a parabola in at most R points. (b) Find the smallest R with this property.

AMM 10462.956. (I.Rivin) Let and be nondegenerate simplices in En,with(n − − 1) dimensional faces Fi and Fi respectively, (i =0, 1, 2,...,n). Let αij be the dihedral angle between Fi and Fj,andletαij be the dihedral angle between Fi and Fj,(i = j). Prove that if ≥ ≤ ≤ αij αij with 0 i

AMM 10469.957.S989*. (J.Anglesio) Let P be a point in the interior of the trianlge ABC and let the lines AP , BP, CP meet the sides BC, CA, AB respectively at the points D, E, F . Let the circles on diameters BC and AD intersect at the points a, a;thecircles on diameters CA and BE intersect at points b, b; and the circles on diameters AB and CF intersect at points c, c. Show that a, a, b, b, c, c lie on a circle.

AMM 10472.957. (E.Kitchen) Let P0P1P2P3P4 be a convex pentagon that is aﬃnely π equivalent to a regular pentagon. Let Lj be the center of a rotation through + 5 radians taking − 3π Pj+2 to Pj−2 (all subscripts modulo 5). Show that Pj is the center of a rotation through 5 radians taking Lj−1 to Lj+1. YIU : Problems in Elementary Geometry 174

AMM 10478.958. (J.P.Hutchison) Let P be a simple closed n−gon, not necessarily convex (an “art galley”), with some pairs of vertices joined by nonintersecting interior diagonals (“walls”), and suppose that in the interior of each of these diagonals there is an arbitrarily placed, arbitrarily small opening (a “doorway”). Determine the size of the smallest set G of points (“guards”) so that every other point q in P there is a line segment in P , disjoint from the punctured diagonals, that joins q to a point of G.

Proposal for AMM Let X, Y, Z be the projections of the incenter of ABC on the sides BC,CA and AB respectively. Let X,Y,Z be the points on the incircle diametrically opposite to X, Y, Z respectively. Show that the lines AX,BY,CZ are concurrent. Solution. The line AX, when produced, intersects the side BC at the point of contact with the excircle on this side. Similarly, for BY and CZ. It follows that these three lines intersect at the Nagel point of the triangle. To justify the ﬁrst statement, let P be the point on BC so that BP = s − c and CP = s − b. This is the point of contact of the side BC with the excircle. Let I be the incenter, Ia the excenter on the side BC,andra the corresponding exradius. Produce XI to meet AP at X . It follows that IX IA r = = . IaP IaA ra Since IaP = ra,wehaveIX = r and X = X , the point on the incircle diametrically opposite to X. With this observation, the concurrency of the lines AX,BY,CZ follows from Ceva’s theorem.

AMM 10474.958. (H.Tamvikas) Consider a triangle ABC and a point P in the interior of ABC, and let the lines AP , BP, CP meet the lines BC, CA, AB at the points D, E, F respectively. Show that EDF is a right angle if and only if 1 1 1 1 = + + . |DP| |AD| |BD| |CD|

Solution. We ﬁrst establish the equivalence of the following statements. (i) DE bisects ADC. (ii) DF bisects ADB. (iii) EDF is a right angle.

|CE| |CD| Suppose DE bisects ADC, so that in triangle ADC, |AE| = |AD| . Applying Menelau’s theorem to triangle ADC with transversal BPE, we have |CE| |AP | |BD| · · =1. |AE| |DP| |BC| YIU : Problems in Elementary Geometry 175

This means |BC| |AP | |CE| |AP | |CD| = · = · . (2) |BD| |DP| |AE| |DP| |AD| This can be reorganized as |BC| |AP | |BD| = · , (3) |CD| |DP| |AD|

|BF| |BD| from which we infer |AF | = |AD| ,andDF bisects ADB. This proves that (i) implies (ii). Similarly, (ii) implies (i). The equivalence of (i) and (ii) clearly shows that each one of them implies (iii). That (iii) implies (i) and (ii) is a consequence of the more general fact: in triangle |CE| ADC, CDE is greater than, equal to, or less than ADE according as |AE| is greater than, |CD| equal to, or less than |AD| . Also, in triangle ADB, BDF is greater than, equal to, or less than |BF| |BD| ADF according as |AF | is greater than, equal to, or less than |AD| . Now, suppose any of (i), (ii), (iii) holds. Then from (1) and (2), we have |BC| |BC| |AP | |CD| |BD| |AP | |BC| + = + = · . |BD| |CD| |DP| |AD| |AD| |DP| |AD|

Consequently, 1 1 |AD|−|DP| 1 1 + = = − , |BD| |CD| |DP|·|AD| |DP| |AD| and 1 1 1 1 = + + . (4) |DP| |AD| |BD| |CD| Conversely, if (3) holds, then

|BC| |AP | = . |CD|·|BD| |DP|·|AD| so that |CD| |BC| |DP| |CE| = · = . |AD| |BD| |AP | |AE| This means that DE bisects ADC.Consequently,DF bisects ADB,and DEF is a right angle.

AMM 10483.959. (S.Rabinowitz) Given an odd positive integer n,letA1, A2, ..., An be a regular n−gon with circumcircle Γ. A circle Oi with radius r is drawn externally tangent to Γ at Ai for i =1, 2,...,n.LetP be any point on Γ between An and A1.AcircleC (with YIU : Problems in Elementary Geometry 176

any radius) is drawn externally tangent to Γ at P .Letti be the length of the common external tangent between the circles C and Oi.Provethat n i (−1) ti =0. i=1

AMM 10514.963.S978. (J.Fukuta) Let ABC,letP1 and P2, P3 and P4, P5 and P6 be thepoints on the sides BC, CA, AB respectively, such that

|BP | |CP | |CP | |AP | |AP | |BP | 1 = 2 = 3 = 4 = 5 = 6 = r, |P1C| |P2B| |P3A| |P4A| |P5B| |P6A| with 0 ≤ r ≤ 1. Let A , B , C be the points of intersections of P1P4 and P2P5, P3P6 and P4P1, P5P2 and P6P3, respectively. Let QiPiPi+1, i =1,...,6 be the equilateral triangles built outwards on the sides of the hexagon P1P2 ···P6.LetRiQi−1Qi+1, i =1, 2,...,6 be the equilateral triangles built outwards on the diagaonals of the hexagon Q1Q2 ···Q6. (a) Show that the points Q1, A and Q4 lie on R1R4. (b) Show that the diagonals R1R4, R2R5 and R3R6 are concurrent, equal in length, and that the angle of intersection of two of these is 60◦. (c) Let Gi be the centroid of the triangle Ri−1RiRi+1, i =1,...,6. Show that G1G2 ···G6 is a regular hexagon and that its center coincides with the centroid of the triangle ABC.

AMM 10517.964.S979. (J.Anglesio) Let ABC be a triangle and√ let H be its orthocenter and I its incenter. If W is the point such that HW =4HI an R =2 2|HI|, show that none of the vertices A, B,orC is in the interior of the circle with center W and radius R.

AMM 10533.966. (A.Flores) On a parallelogram P construct exterior squares on the sides. The centers of these squares form a square QE. On the same parallelogram construct the interior squares on the sides. The centers of these squares form another square QI . (a) Show that area (QE)-area(QI )=2area(P ). (b) Is there a generalization when P is replaced by an arbitrary convex quadrilateral?

AMM 10542.967.S982. (J.Anglesio) Let C be the circumcircle of a trinangle A0B0C0 and I the incircle. It is known that, for each point A on C, there is a triangle ABC having C for circumcircle and I for incircle. Show that the locus of the centroid G of triangle ABC is a circle that is traversed three times by G as A traverses C once, and determine the center and radius of this circle. YIU : Problems in Elementary Geometry 177

AMM 10547.968. (D.Sachelarie and V.Sachelarie) In the triangle ABC,letO be the circumcenter, H the orthocenter, and I the incenter. Prove that the triangle OHI is isosceles if and only if a3 + b3 + c3 R = . 3abc 2r

AMM 10560.9610.S982. (E.Alkan) Consider a convex quadrilateral ABCD,andchoose points P , Q, R,andS on sides AB, BC, CD, DA respectively, with

|PA| |RD| |QB| |SA| = and = . |PB| |RC| |QC| |SD|

Let K denote the area of ABCD,andletKA, KB, KC and KD denote the areas of SAP, PBQ, 4 12 QCR, RDS respectively. Show that K ≥ 2 KAKBKC KD and determine a necessary and suﬃcient condition for equality.

AMM 10602.976.S985. (Dan Sachelarie and Vlad Sachelarie) In triangle ABC,prove that π 2r HIO ≥ +arcsin , 2 R π with equality if and only if HIN = 2 .

AMM 10616.978. (Ernesto Bruno Cossi) Let K ba a compact, covex set in the plane. ForeachinteriorpointP of K and each line through P ,letA and B be the two points of on the boundary of K,andletQ be the harmonic conjugate of P with respect to A and B.If K is an ellipse, then for each P the locus of points Q is a straight line. Is the converse true?

AMM 10631.9710. (Greg Huber) Given a triangle T , let the intriangle of T be the triangle whose vertices are the points where the circle inscribed in T touches T . Given a triangle T0, form a sequence of triangles T0, T1, T2,...,inwhicheachTn+1 is the intriangle of Tn.Let dn+1 →∞ dn be the distance between the incenters of Tn and Tn+1. Find limn dn when T0 is not equilateral. YIU : Problems in Elementary Geometry 178

AMM 10637.981.S998. (C.F.Parry) Suppose triangle ABC has circumcircle Γ, circumcenter O, and orthocenter H. Parallel lines α, β, γ are drawn through the vertices A, B, C respectively. Let α, β, γ be the reﬂections of α, β, γ in the sides BC, CA, AB respectively. (a) Show that α, β, γ are concurrent if and only if α, β, γ are parallel to the Euler line OH. (b) Suppose that α, β, γ are concurrent at the point P . Show that Γ bisects OP.

AMM 10644.982.S995. (Mih`aly Bencze) Given an acute triangle ABC with sides of length a, b, c,inradiusr, and circumradius R, prove that r abc ≤ . 2R 2(a2 + b2)(b2 + c2)(c2 + a2)

AMM 10653.983.S00(1)89–91. (Marcin Mazur) Given an isosceles triangle ABC, prove that there is a unique set of points A1, B1, C1 on sides BC, CA, AB respectively, with the property that the quadrilaterals AC1A1B1, BA1B1C1, CB1C1A1 circumscribe circles. Furthermore, prove that the inradius of ABC is twice the inradius of A1B1C1.

AMM 10659.984.S999. (Jiro Fukuta) Let D, E, F be points in the interior of sides BC, CA, AB respectively, of triangle ABC such that the incircles of AEF , BFD,andCDE are congruent, each having radius r.Let ρ, s, K be the inradius, semiperimeter, and area of triangle ABC,andletρ, s, K be the corresponding quantities for DEF. − − r − r 2 (a) Prove that ρ = ρ r, s =(1 ρ )s,andK =(1 ρ ) K. ρ (b) Prove that, if r = 2 ,thenD, E, F are midpoints of the sides of triangle ABC. See also Crux 1191.

AMM 10662.985.S999. (J. Konhauser and Stan Wagon) Find a construction for the center of gravity of the edges of a quadrilateral.

AMM 10673.98(6)559.S00(2)180–181. (Marcin Mazur) Let C be the circle insrcibed in the triangle A1A2A3,andletPi ∈ Ai+1Ai+2 (subscripts taken modulo 3) be such that the lines PiAi are concurrent. Let ti be the second tangent from Pi to C, the ﬁrst being Ai+1Ai+2. Prove that the points Q1, Q2, Q3 deﬁned by Qi = ti ∩ Pi+1Pi+2 are collinear.

AMM 10678.98(7)666.S00(4)373. (Kimberling and Yﬀ) Let C be the incircle of YIU : Problems in Elementary Geometry 179

triangle A1A2A3. Suppose that whenever {i, j, k} = {1, 2, 3}, there is a circle through Aj and Ak meeting C in a single point B − i. Prove that the lines A1B1, A2B2, A3B3 are concurrent.

AMM 10686.98(8)768.S00(7)656–657. (C.R.Pranesachar) An equicevian point of a triangle ABC is a point P (not necessarily inside the triangle) such that the cevians on the lines AP , BP, CP have equal length. Let SBC be an equilateral triangle, and let A be chosen in the interior of SBC on the altitude dropped from S. (a) Show that ABC has two equicevian points. (b) Show that the common length of the cevians through either of the equicevian points is constant, independent of the choice of A. (c) Show that the equicevian points divide the cevian through A in a constant ratio, independent of the choice of A. (d) Find the locus of the equicevian points as A varies. (e) Let S be the reﬂection of S in the line BC. Show that (a), (b), and (c) hold if A moves on any ellipse with S and S as foci. Find the locus of the equicevian points as A varies on the ellipse.

AMM 10693.98(9)859.S00(2)182–184. (Wu Wei Chao) Let P be an arbitrary point on the side BC of triangle ABC. (a) Let D be the point where the incircle of triangle ABC meets BC,andletQ and R be the incenters of ABP and ACP respectively. Prove that QDR is a right angle. (b) Let D be the point where the excircle opposite A of ABC meets BC.LetQ and R be the excenters opposite A of ABP and ACP respectively. Prove that QDR ∼RDQ. (c) Prove that the lines BC, QR, QR are concurrent or parallel.

AMM 10698.98(10)995.S00(7)657–658. (Wu Wei Chao) Let P be the intersection of the two diagonals of a convex quadrilateral ABCD. Let the radii of the circles inscribed in the four triangles AP B, BPC, CPD,andDPA be r1, r2, r3 and r4 respectively. Show that ABCD has an inscribed circle if and only if 1 1 1 1 + = + . r1 r3 r2 r4

AMM 10703.98(10)956.S00(3)285. (J.Angelsio) Given triangle XY Z, let its incenter be I, its centroid C, its circumcenter O, its orthocenter H, the center of its nine-point circle W , its Gergonne point G, its Nagel point N.LetS denote the intersection of the line IG with the Euler, and let T , U, V denote respectively the intersections of the line IG with lines NO, NW, and NH. (a) Show that C lies one third of the way from H to S (so that SO = HO). YIU : Problems in Elementary Geometry 180

(b) Show that ST : SI : SU : SV = 10 : 15 : 18 : 30. (c) Show that NO : TO =3:1;NW : UW =5:3,andNH : VH.[Wemaynowinferthat NH =2· OI and that these segments are parallel].

AMM 10704.991. (Wiliam G Spohn) Show that there are inﬁnitely many pairs ((a, b, c), (a,b,c)) of primitive Pythagorean triples such that |a − a|, |b − b|,and|c − c| areallequalto3or4. Examples include ((12,5,13), (15,8,17)) and ((77,36,85), (80,30,89)).

AMM 10710.99(1)68.S00(6)572–573. (Bogdan Suceava) Let ABC be an acute triangle with incenter I,andletD, E,andF be the points where the circle inscribed in ABC touches BC, CA,andAB respectively. Let M be the intersection of the line[s] through A parallel to BC and DE,andletN be the intersection of the line[s] through A parallel to BC and DF.Let P and Q be the midpoints of DM and DN respectively. Prove that A, E, F , I, P , Q are on the same circle.

AMM 10713.99(2)166.S00(5)464. (Juan - Bosco Romero M´arquez) Given a triangle with angles A ≥ B ≥ C,leta, b,andc be the lengths of the corresponding opposite sides, let r be the radius of the inscribed circle, and let R the the radius of the circumscribed circle. Show 1 that A is acute if and only if R + r<2 (b + c).

AMM 10717.99(2)167.S00(5)466-167. (M.Mazur) We say that a tetrahedron is rigid if it is determined by its volume, the areas of its faces, and the radius of its circumscribed sphere. We say that a tetrahedron is very rigid if it is determined just by the areas of its faces and the radius of its circumscribed sphere. (a) Prove that every tetrahedron with faces of equal area is rigid. (b) Prove that a very rigid tetrahedron with faces of equal area is regular. (c)* Is every tetrahedron rigid? (d)* Is every very tetrahedron regular?

AMM 10719.99(3)264.S00(10)952–954. (Jean Anglesio) Let A, I,andG be three points in the plane. Let M denote the point 2/3ofthewayfromA to I,andletU and V be the circles of radius |AM| each of which is tangent to AI at M. Show that when G is outside both U and V , there are precisely two triangles ABC with incenter I and centroid G.Provide a Euclidean construction for them. Show that when G is in the interior of U or V , there does not exist a triangle ABC with incenter I and centroid G. Solution: Consider a triangle ABC with incenter I and centroid G.LetD be the midpoint of 1 the side BC. Clearly, GD = 2 AG.LetX be the orthogonal projection of I on BC. This is the point of tangency of the incircle with side BC.NotethatX lies on the circle with diameter ID. Let Y be the point on BC, symmetric to X with respect to D. This is the point of tangency of YIU : Problems in Elementary Geometry 181

BC with the excircle on this side. The segment AY passes through a point N, called the Nagel point of the triangle, which lies on the line IG,withGN =2IG. To construct triangle ABC given a vertex A, the incenter I, and the centroid G, we therefore proceed in the following steps. (1) Locate point D which divides the segment AG externally in the ratio 3 : −2. (2) Locate point N which divides the segment IG externally in the ratio IN : NG =3:−2. Note that AN and ID are parallel. (3) Construct the circle C with ID as diameter. (4) Extend the segment ND to N such that ND = DN, and construct a line through N parallel to AN. The intersections of with C, if any, are the possible locations of the point X, the point of tangency of the incircle with the side BC. The triangle ABC is then bounded by the line DX and the two tangents from A to the circle which has center I and passes through X.

It remains to determine the condition for real intersections of and C. Obviously, a necessary and suﬃcient condition is that N lies on or inside the circle C. See MG 1149.824, and W. Wernick, Triangle constructions with three located points, Math. Mag. 55 (1982) 227 – 230. This is also a follow-upof this article by Meyers around 1995.

AMM 10734.99(4)470.S00(7)658–659. (Floor van Lamoen) Let ABC be a triangle with orthocenter H, incenter I, and circumcenter O.Let[P, r] denote the circle with center P and radius r. Show that the radical center of [A, CA + AB], [B,AB + BC], and [C, BC + CA] is the point obtained by reﬂecting H through O and then reﬂecting the result through I. This is the same as CMJ 664.995 (J. Fukuta).

AMM 10749.997. (Alain Grigis) Let ABC be a triangle with a right angle at B and an π angle of 6 at A. Consider a billiard path in the triangle that begins at A, reflects sucessively off side BC at P ,offsideAB at R,offsideAC at S, and then ends at B. (a) Show that AP , QR,andSB are concurrent at a point X. π (b) Show that the angles formed at X measures 3 . (c) Show that AX = XP + PQ+ QX = XR + RS + SX =2XB.

AMM 10755.998. (Jiro Fukuta) An arbitrary circle O is drawn through vertices B and D of a convex quadrilateral ABCD.LetO1 be the circle tangent to lines AB and AD and tangent to O internally at a point of O on the opposite side of line BD from A.LetO2 be the circle tangent to lines CB and CD and tangent to O internally at a point of O on the opposite side of line BD from C.LetR1 and R2 be the radii of circles O1 and O2, respectively, and let YIU : Problems in Elementary Geometry 182

r1 and r2 be the radii of the incircles of triangles ABD and CBD respectively. Prove that the r1 r2 quadrilateral ABCD is inscribable in a cicle if and only if R1 + R2 =1.

AMM 10756.998. (Douglas Iannucci) Prove that √ π 1 7 1 1 √ 1 1 cos = + (cos( arccos √ )+ 3sin( arccos √ )). 7 6 6 3 2 7 3 2 7

AMM 10759.99(8)778.S00(9)867–868. (Calin Popescu) In triangle ABC,letha denote the altitude to the side BC and let ra denote the exradius relative to the side BC,etc.Prove that n n n n n n ≤ n n n n n n ha ra + hb rb + hc rc ra rb + rb rc + rc ra .

AMM 10763.999. (J. Angelsio) Let ABC be a triangle; let O be its circumcenter, H its orthocenter, I its incenter, N its Nagel point, and X, Y , Z its excenters. Let S be deﬁned so that O is the midpoint of HS,andletT denote the midpoint of SN. It is known that the orthocenter and the nine-point center of triangle XY Z are I and O, respectively. Prove that (a) the circumcenter of triangle is T ;and (b) the centroid of triangle XY Z is the centroid of SIN.

AMM 10780.00(1)84.S00(10)957–158. (K. Kedlaya) Let T be a triangle. Two circles in T are called partners if they are the incircles of two triangles with disjoint interior whose union is T . Every circle tangent exactly to two sides of T has two partners. Let C1, C2, ..., C6 be distinct circles such that Ci, Ci+1 are partners for each i ∈{1, 2, 3, 4, 5, 6}. Show that C6 and C1 are partners. See E3236.

AMM 10783.00(2)176. (Wu Wei Chao) Let ABCD be a cyclic quadrilateral such that AD is not parallel to BC.GivenpointsE and F on the line CD,letG and H be the circumcenters of BCE and ADF . Prove that the lines AB, CD,andGH are concurrent or parallel if and only if there is a circle through A, B, E, F .

AMM 10796.00(4)367. (Floor van Lamoen) Let ABC be a triangle and let the feet of the altitudes dropped from A, B, C be A, B, C respectively. Show that the Euler lines of triangles ABC, ABC,andABC concur at a point on the nine-point circle of ABC.

AMM 10804.00(5)462. (A. Sinefakopoulos) Let ABCD be a convex quadrilateral with an incircle that contacts AB at E and CD at F .SHowthatABCD has a circumcircle if and AE DF only if EB = FC. YIU : Problems in Elementary Geometry 183

AMM10810.00(6)566. (J-B. Romero M´arquez) Consider a convex quadrilateral with no parallel sides. On each side AB select a pint T as follows: Draw lines from A and B parallel to the opposite side. Let A and B be the new points where these lines intersect the sides neighbouring AB.LetT be the point where AB intersects AB. Prove that the four points selected in this way are the corners of a parallelogram.

AMM 10814.00(6)567. (R.Satnoianu) Let P be in the interior of triangle ABC.Letr, s, t be the distances from P to the vertices A, B, C respectively, and let x, y, z be the distances from P to the sides BC, CA, AB respectively. Prove that (a) qr + qs + qt +3≥ 2(qx + qy + qz) for any q ≥ 1, (b) qs+t + qt+r + qr+s +6≥ q2x + q2y + q2z +2(qx + qy + qz) for any qge1.

AMM 10830.00(9)863. (Floor van Lamoen) A triangle is divided by its three medians into 6 smaller triangles. Show that the circumcenters of these smaller triangles lie on a circle. See also VIS 274. The center is the point

[10a4 − 13a2(b2 + c2)+2(2b4 − 5b2c2 +2c4)].

Check!

AMM 10838.00(10)950. (F.S.Pˆarvˆanescu) Let M be any point in the interior of triangle ABC,andletD, E, F be points on the perimeter such that AD, BE, CF are concurrent at M. Show that if the triangles BMD, CME, AMF all have equal areas and equal perimeters, then ABC is equilateral.

AMM 10845.01(1)77. (C. Demeter) Let E be a point inside the triangle ABC such that ABE = ACE.LetF and G be the feet of the perpendiculars from E to the internal and external bisectors, respectively, of angle BAC. Prove that the line FG passes through the midpoint of BC.

AMM 10862.01(1)271. (Wu Wei Chao) Let P be a point in the interior of triangle ABC. Prove that PCB = PAC = PBC if and only if BP ·AC = CP ·AB amd BPC + BAC = 180◦. YIU : Problems in Elementary Geometry 184

Mathematics Magazine 1976 – 1996

MG867.733.S742.(Carlitz) Let P be a point in the interior of ABC.LetR1,R2,R3 denote the distances of P from the vertices of ABC and let r1,r2,r3 denote the distances from P to the sides of ABC. Show that ≥ 2 ≥ 2 2 ≥ 2 2 2 r1R2R3 12r1r2r3; r1R1 12r1r2r3; r2r3R2R3 12r1r2r3. In each case there is equality if and only if ABC is equilateral and P is the center of ABC.

MG880.735.S751.(R.Corry) Given chords a, b, c in a semicircle. Determine the diameter of the circle. Solution. D3 − (a2 + b2 + c2)D − 2abc =0. See also AMME574.435.

MG896.742.(S.B.Maurer) Prove that there are inﬁnitely many Pythagorean triplets of the form (a, a +1,c).

MG898.742.(R.D.H.Jones) There are ﬁve points associated with every triangle: the orthocenter, the centroid, the incenter, the circumcenter, and the nine-point center. Prove that if any two of these coincide the triangle is equilateral.

MG900.742.(Klamkin) A long sheet of rectangular paper ABCD is folded such that D falls on AB producing a smooth crease EF with E on AD and F on CD when unfolded. Determine the minimal area of triangle EFD by elementary methods.

MG901.743.S753.(Bankoﬀ) The sides of any triangle are rational or integral only if the ratio of the inradius to the circumradius is rational. Is the converse true ? YIU : Problems in Elementary Geometry 185

Answer. The converse is not true. M.Goldberg suggested that the problem could have been stated as follows: If the sides a, b, c have rational ratios, then the ratio of the inradius r to the abc circumradius R is rational. If K is the area of the triangle and s its semiperimeter, then R = 4K K R (sic!) and r = s .Giventhefactthatr is rational, it does not follow that the ratios of the sides of the triangle are always rational.

MG910.744.(L.Carlitz) Let P be a point in the interior of ABC and let r1,r2,r3 denote the distances from P to the sides of ABC.Leta, b, c denote the sides and r the radius of the incircle of ABC. Show that

a b c 2s + + ≥ , r1 r2 r3 r 2 2 2 ≥ 2 ar1 + br2 + cr3 2r s, 2 (s − a)r2r3 +(s − b)r1r3 +(s − c)r1r2 ≤ r s, 2 2 2 − − − ≥ 2 ar1 + br2 + cr3 +(s a)r2r3 +(s b)r3r1 +(s c)r1r2 3r s, where 2s = a + b + c. In each case there is equality if and only if P is the incenter of ABC.

MG913.744.(Garfunkel) Triangles A1B1C1 is inscribed in a circle. The medians are drawn and extended to the circle meeting the circle at points A2B2C2. The medians of triangle A2B2C2 are likewise drawn and extended to the circle to points A3B3C3, and so on. Prove that triangle AnBnCn becomes equilateral as n →∞(and very rapidly).

MG916.745.(H.Demir) Let XY Z be the pedal triangle of a point P with regard to the triangle ABC. Then ﬁnd the trilinear coordinates x, y, z of P such that YA+AZ = ZB+BX = XC + CY .

MG917.745.(Trigg) The length of every edge of a regular pentagonal prism is e. (a) When the two pentagonal faces are rotated about parallel diagonals until two of their edges coincide, two lateral edges vanish and one becomes elongated. The resulting hexahedron has two congruent regular pentagons, two congruent equilateral triangles, and two congruent trapezoids for faces. Eleven of its edges are equal. What is the length of the twelfth edge ? (b) When the pentagonal faces are otherwise rotated about the parallel diagonals until two of the vertices coincide, one lateral edge vanishes and two are elongated. The resulting heptahedrron has two congruent regular pentagons, two congruent equilateral triangles, two congruent trapezoids and one rectangle for faces. Twelve of its edges are equal. What are the lengths of the other two edges ?

MG919.745.(Klamkin) An (n+1)−dimensional simplex with vertices O, A1, A2,...,An+1 is such that the (n+1) congruent edges OAi are mutually orthogonal. Show that the orthogonal YIU : Problems in Elementary Geometry 186 projection of O onto the n−dimensional face opposite to it coincides with the orthocenter of that face. (Thsi generalizes the known result for n =2.

rr1r2r3 MG920.(Bankoﬀ) The radius of the nine-point circle is equal to h1h2h3 .

MG925.751. (a) (J.G.Baron) Prove that any non-self-intersecting cyclic octagon is such that the sum of any four nonadjacent interior angles is 3π. (b) (T.E.Elsner) An octagon is inscribed in a circle with vertices on any four diameters. Show that each alternate pair of exterior angles is complementary.

MG929.752.(Trigg) Show that there are two octahedra with equilateral triangular faces. √ MG936.752.(Garfunkel) It is√ known that ha + hb + hc ≤ 3s. Prove or disprove the stronger inequality ta + tb + mc ≤ 3s.

MG945.754.S764.(A.Wayne) Find the smallest Pythagorean triangle in which a square with integer sides can be inscribed so that an angle of the square coincides with the right angle of the triangle. Solution. Consider a right triangle whose√ sides are (ka, kb, kc), with a, b, c relatively prime. · ab The bisector of the right angle has length k 2 a+b . Each side of the square insribed in the · ab triangle has rational length k a+b . To make this an integer, we take k = a + b. The area of the 1 2 triangle is 2 ab(a + b) . This is smallest when a =3,b =4,a + b = 7. The triangle has sides 21, 28, 35, and the inscribed square has side 12.

MG949.754.S764.(Erd¨os) In a circle with center O, OXY is perpendicular to chord AB (as shown, MG(1975)p.239). Prove that DX ≤ CY .

MG955.755.(C.F.White) For three line segments of unequal lengths a, b,andc drawn on a plane from a common point, characterize the proper angular positions such that the outer end-points of the line segments deﬁne the maximum-area triangle. Show how to approximate the exact values of the edges for a =3,b =4andc =5. YIU : Problems in Elementary Geometry 187

MG959.755.(Carlitz) Let P be a point in the interior of ABC and let r1,r2,r3 denote the distances from P to the sides of ABC.LetR denote the circumradius. Show that √ √ √ R r + r + r ≤ 3 . 1 2 3 2

MG960.755.(A.Wayne) In a rectangle of dimensions a and b, lines parallel to the sides divide the interior into ab square unit areas. Through the interior of how many of these units squares will a diagonal of the rectangle pass?

MG761.p25. Niven, A new proof of Routh’s theorem: If the sides BC, CA, AB of ABC are divided at points D, E, F in the ratios 1 : λ, 1:µ, 1:ν respectively, then the area of the triangle formed by AD, BE, CF is (λµν − 1)2 . (λµ + µ +1)(µν + ν +1)(νλ + λ +1) See also CM7.p199,273, Three more proofs of Routh’s theorem.

MG963.761.S771. Characterize convex quadrilaterals with sides a, b, c, d such that the determinant of the cyclic matrix with entries a, b, c, d is zero. Solution. Either a + c = b + d or a = c, b = d. In the ﬁrst case, the quadrilateral can be circumscribed about a circle. In the

MG966.761.S773. ApointP lies in the interior of rectangle of sides a and b.(i)Finda, b and P so that all eight distances from P to the four vertices and the four sides are positive integers. (ii) Find an example of a square where seven of the distances are integers. (iii) Can all eight distances be a square for a square? See also MG813.p131, MG1147.823. MG865.73?, GuyD19..

MG967.761.S773. Let ABC be a triangle inscribed in a circle with internal bisectors of B and C meeting the circle again in the points B1 and C1 respectively. (i) If B = C,thenBB1 = CC1. (ii) Characterize ABC for which BB1 = CC1. Solution. A = 60 degrees, or B = C.

MG988.764.S781. A given equilateral triangle ABC is projected orthogonally from a given plane P to another plane P . Show that the sum of the squares of the sides of ABC is independent of the orientation of ABC in the plane P . YIU : Problems in Elementary Geometry 188

MG998.765.S783. Characterize all triangle in which the the triangle whose vertices are the feet of the internal angle bisectors is a right angle. Solution. ACB is a right angle if and only if ACB = 120 degrees.

MG1010.773.S795. Given ABC,pointsD, E, F are on the lines determined by BC, CA, AB respectively. Assume AD = BE = CF. The lines AD, BE, CF intersect to form PQR. (i) Show that PQR is equilateral if and only if ABC is. (ii) Express the area of PQR in terms of that of ABC.

MG1020.773.S791. For i =1, 2, 3, let the circle Ci have center (hi,ki) and radius ri.Find a determinant equation for the circle orthogonal to these three given circles.

MG1023.774.S791. Call a triangle super-Heronian if it has integral sides and integral area, and the sides are consecutive integers. Are there inﬁnitely many distinct super-Heronian triangles? − a 2 − Solution. With sides a 1,a,a +1, = 4 3(a 4). is an integer if and only if 1 2 − 2 − 4 3(a 4) is an integer. Now, an+1 = an 2witha1 = 14 deﬁnes a recurrent sequence 1 2 − such that 4 3(an 4) is always an integer. These gives distinct, nonsimilar super-Heronian triangles. (13, 14, 15; 84).

MG775.p261 Klamkin, An ellipse inequality.

MG1028.775.S793 Let P1,P2,P3 be arbitrary points in the plane of ABC. Let arbitrary lines perpendicualr to APi,BPi,CPi determine AiBiCi,i=1, 2, 3. Now, let A0,B0,C0 be the respective centroids of triangles A1A2A3,B1B2B3,C1C2C3. Show that the perpendiculars from A, B, C on the sides of A0B0C0 concur. ∞ 1 kπ kπ MG1039.783.S794 Sum the series k=1 k2 tan m tan n .

MG1043.783.S795 For two triangles,

s1 s2 1 1 1 1 { } 2 ≥ 3{√ + √ + √ }, r1R1 r2R2 a1a2 b1b2 c1c2 with equality if and only if the triangles are equilateral. Also show that the analogous three triangle inequality s s s 1 1 1 1 2 3 ≥ 9{√ + √ + √ } r1R1 r2R2 r3R3 a1a2a3 b1b2b3 c1c2c3 is not valid. YIU : Problems in Elementary Geometry 189

MG1054.785.S801 Construct ABC by straightedge and compass given (i) a, ma,wa. (ii) A, ma,wa. Solution by Howard Eves, with an interesting quotation.

MG1057.785.S802. Disseect a regular pentagon into six pieces and reassemble the pieces to form three regular pentagons whose sides are in the ratio 2 : 2 : 1.

MG1076.794.S804.(Klamkin) Let B be an n-gon inscribed in a regular n-gon A. Show that the vectices of B divide each side of A in the same ratio and sense if and only if B is regular. See also CMJ794.146.S811, CMJ815.203.

MG1077.794.S804. Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is pm,wherep is a prime and m a positive integer, is m +1,if p =2, and 2m +1,ifp =2. See also MG795.1088.S811, CMJ845.p429, Crux 2012.

MG1088.795.S811. (a) For each positive m, how many Pythagorean triangles are there whichhaveanareaequaltom times the perimeter? How many of these are primitive? (b)* Can this result be generalized to all triangles with integer sides and area equal to m times the perimeter? See also MG1077.794.S804, CMJ845.p429 where (a) appears as a conjecture. Write m = k0 k1 ··· kr 2 p1 pr as a product of distinct prime powers. The number of Pythagorean triangles with r area m times perimeters is 2 ,amongthese(k0 + 2)(2k1 +1)···(2kr + 1) are primitive. Steven Kleiman and Klostergaard have written an article to discuss an algorithm for solving part (b).

MG1107.805.S821. Determine the maximum value of

sin A1 sin A2 ···sin An given tan A1 tan A2 ···tan An =1.

MG1119.812.S823. Let ABC be inscribed in a circle and let point P be the centroid of the triangle. The line segments AP, BP, CP are extended to meet the circle in points D, E, F respectively. Prove that AP BP CP + + =3. PD PE PF YIU : Problems in Elementary Geometry 190

MG1120.812.S823. Let ABC be inscribed in a circle and let P be a point in the interior of the triangle. The line segments AP, BP, CP are extended to meet the circle in points D, E, F respectively. Describe all points P for which AP BP CP + + ≤ 3. PD PE PF See also Crux 643.S8.155.

MG1129.814.S825 Find radii r, R,withr

MG1132.815.S831. Let AD, BE, CF be cevians of ABC intersecting at P . (i) Show that if AD bisects angle A and BD · CE = DC · BF,thenABC is an isosceles triangle. (ii) Show that if AD, BE, CF are bisectors and BP · FP = BF · AP ,thenABC is a right triangle.

MG1147.823.S833. If O is a point in the interior of rectangle ABCD and OA = a, OB = b, OC = c,whatisOD? Given one such triple, what is the maximum area of the rectangle? See also MG966.761.S773, MG813.p131, AMME3208.

MG1149.824.S841 Construct ABC given (i) O, Ma,I. (ii) O, Ha,Ta. (iii) Ma,Ha,I.

MG1151.824.S834 Three points P, Q, R move on curves in the plane. At each instant, the normal at P to the curve on which P is moving coincides with the bisector of the angle RPQ. Corresponding conditions hold for the points Q and R. (i) Show that the perimeter of the triangle PQR is constant. (ii) Find examples other than that of an equilateral triangle whose vertices move around a ﬁxed circle. (iii) Does the result in (i) have a dynamic interpretation in terms of three heavy masses moving on a smooth horizontal table with a light inextensible string looped over them?

MG1155.825.S841 A plane intersects a sphere forming two spherical segments. Let S be one of these segments and let A be the point of the sphere furthest from the segment S.Prove that the length of the tangaent from A to a variable sphere inscribed in the segment S is a constant. YIU : Problems in Elementary Geometry 191

√ MG1156.825.S841 a(s − a)+b(s − b)+c(s − c) ≥ 12, with equality if and only both triangles are equilateral.

MG1157.825.S841 The interior surface of a wine glass is a right circular cone. The glass contains some wine and is tilted so that the wine-to-air interface is an ellipse of eccentricity e and is at right angles to a generator of the cone. Prove that the area of the ellipse is e times the area of that part of the curved surface of the cone which is in contact with the wine.

MG1161.831.S854 Two equilateral triangles are placed so that their intersection is a hexagon (not necessarily regular). The vertices of the equilateral trianlges are connected to form an outer hexagon. Show that if three alternate angles of the outer hexagon are equal, then the triangles have the same center. See also Crux 745.

MG1168.832.S842 Let P be a variable point on side BC of ABC.

MG1170.833.S843 In ABC, the bisectors of the angles are the segments AP, BQ, CR.If a =4,b =5,c = 6, ﬁnd the size of angle QP R.

MG1181.835.S845 The cevians AD, BE, CF of ABC intersect at P .Iftheareasof BDP,CEP and AF P are equal, then P is the centroid G.

MG1187.842.S852 (A three winged butterﬂy problem) Let the chord AB of circle O be trisected at C and D.LetP be any point on the circle other than A and B. Extend the lines PD and PC to intersect the circle in E and F respectively. Extend the lines EC and FD to intersect the circle in G and H respectively. Let GH and HE intersect AB in L and M respectively. Prove that AL = BM.

MG1191.843.S853 Let P1,P2,...,Pn be the vertices of a regular polygon inscribed in the unite circle. Let An denote the sum, and Bn the arithmetic mean, of the area of all triangles P1PjPk for 1

MG1197.844.S854. Characterize the triangles of which the midpoints of the altitudes are collinear.

MG1199.844. In isosceles ABC with AB = AC,letH be the foot of the altitude from A,andE the foot of the perpendicular from H to AB, M the midpoint of EH. Show that AM ⊥ EC. YIU : Problems in Elementary Geometry 192

Remark. BCE and HAM are similar. n k x 2 x →∞ MG1201.845.S855. Sum Sn = k=0 2 tan 2k tan 2k+1 , and ﬁnd limn Sn.

MG1206.851.S861 Subdivide the side BC of ABC by points B = P0,P1,...,Pn−1,Pn = C in order. If ri is the inradius of APi−1Pi for i =1,...,n,provethat 1 s r + ...+ r < h ln . 1 n 2 a s − a

MG1211.852 Find the locus of points under which an ellipse is seen under a constant angle.

MG1224.854.S864 Consider the nonconvex quadrilateral ABCD in the picture. Let I and J be chosen so that DI = CF and BJ = CE.LetK and L be the points where the line IJ intersects AB and AD respectively. Show that KJ = IL.

MG1230.855.S865 Let ABC and ABC be two similar and similarly oriented triangles in a plane. Let AAA,BBB and CCC be three triangles lying the plane and similar and similarly oriented so ABC.ProvethatABC is similar and similarly oriented to ABC.

MG1232.861.S871 Let l be the Euler line of the nonisosceles triangle ABC and let d be the interal angle bisector of C.Provethat π (i) l is perpendicular to d if and only if γ = 3 . 2π (ii) l is parallel to d if and only if γ = 3 .

MG1236.861.S871 Let the functions f and g be deﬁned by

π2x 8x f(x)= and g(x)= 2π2 +8x2 4π + πx2 for all real x.(a)ProvethatifA, B, C aare the angles of an acute-angled triangles, and R its circumradius, then a + b + c f(A)+f(B)+f(C) <

MG1238.862 (a) Prove the interior of a triangle contains a point P for which the three triangles APB,BPC,CPA have congruent incircles. YIU : Problems in Elementary Geometry 193

(b)* Is P uniquely determined? Can the radii be detemined ? What can you say abolut the property of P ?

MG1256.865.S881 Let ABCD be a cyclic quadrilateral, let the angle bisectors at A and B meet at E, and let the line through E parallel to side CD intersect AD at L and BC at M. Prove that LA + MB = LM.

MG1261.871.S882 (a) What is the area of the smallest triangle with integral sides and integral area? (b)* What is the volume of the smallest tetrahedron with integral sides and inte See also AMME3279.

MG1265.872 Determine the maximum area of a triangle if one sides is of length λ and two of its medians intersect at right angles.

MG1278.875.S885 If O is a given point on the prolongation of diameter BA of a given semicircle, and if ODC is a secant cutting the semicircle in D and C, prove that quadrilateral ABCD has maximum area when the orthogonal projection of DC on AB is equal to the radius of the semicircle.

MG1287.881.S891 Let P be an interior point of the rectangle ABCD. Draw lines through A, B, C, D perpendicular to PA,PB,PC,PD respectively. Show that the area of the convex quadrilateral enclosed by these four lines is equal to or greater than twice the area of the rectangle. When do we have equality?

MG1295.882.S892 Let R be a given rectangle. Construct a square outwards on the length of R; construct another square outwards on the length of the resulting rectangle. Continue this process anticlockwise indeﬁnitely. (a) Prove that the centers of the spiraling squares lie on two perpendicular lines. (b) As the process continues, show that the ratio of the sides of the rectangles approaches the golden mean.

MG1298.883.S893 A quadrilateral ABCD is circumscribed about a circle, and P, Q, R, S are the points of tangency of sides AB,BC,CD,DA respectively. Let a = AB, b = BC,c = ac bd CD,d = DA, p = QS, q = PR. Show that p2 = q2 .

MG1305.884 Let P0 = B,P1,P2,...,Pn = C be points, taken in that order, on the side BC of the triangle ABC.Ifr, ri and h denote respectively the inradii of ABC, APi−1Ai and the YIU : Problems in Elementary Geometry 194 common altitude, prove that n 2r 2r (1 − i )=1− . h h i=1

MG1307.884.S895 Let ABC be a triangle with altitudes ha,hb,hc,andP apointinside or on the boundary of the triangle. Show that 2 PA+ PB + PC ≥ (h + h + h ) 3 a b c with equality if and only if the triangle is equilateral and P is its center.

MG1316.891.S901 Characterize the Heronian triangle in which the Eulerian segment OH subtends a right angle at the vertex A. (A Heronian triangle is one with integer sides and integer area.)

MG1320.892.S902 Let C(I) be a circle with center I,andD, E, F the points of intersection of C(I) with the lines from I perpendicular to the the sides BC, CA, AB respectively. Show that AD, BE, CF are concurrent.

MG1322.893.S902 An n-gon of consecutive sides a1,a2,...,an is circumsrcibed about a circle of unit radius. Determine the minimum value of the products of all its sides.

MG1237.894.S904 Let the sides PQ, QR, RS.SP of a convex quadrangle PQRS touch an inscribed circle at A, B, C, D and let the midpoints of the sides AB,BC,CD,DAbe E,F,G,H. Show that the angle between the diagonals PR,QS is equal to the angle btween the bimedians EG and FH.

MG1333.895.S904 Prove that the quadrilateral formed by the adjacent quadrisectors of the angles of a rhombus is a square.

MG1340.901 Show that the 3 degree angle is the only constructible angle of prime degree measure.

MG1351.903 In the acute triangle ABC,letD be the foot of the perpendicualr from A to BC,letE be the foot of the perpendicular from D to AC and let F be a point on the line segment DE.ProvethatAF is perpendicular to BE if and only if FE/FD = BD/DC.

MG1354.904.S914 Let ABCD be a convex quadrialteral in the plane with trisection points joined as in the ﬁgure to form nine smaller quadrilaterals. YIU : Problems in Elementary Geometry 195

(a) Show that the area of ABCD is one-ninth the area of ABCD. (b) Give necessary and suﬃcient conditions so that all nine quadrilaterals have equal area.

MG1356.904.S914 Let P, Q be points taken on the side BC of ABC, in the order B,P,Q,C. Let the circumcircles of PAB,QAC intersect at M = A and those of PAC,QAB at N. Show that A, M, N are collinear if and only if P and Q are symmetric in the midpoint A of BC.

MG1362.904.S915 1 1 1 1 1 1 − − − si 1 − − − n n n ≤ n ≤ n n n n 3 ( ai + bi + ci ) ( ) 2 (ai + bi + ci ), riRi where the sums and products are over i =1,...,n.

MG1364.911.S921 The incircle of ABC touches BC,CS,AB at points D, E, F respectively. Let P be any point inside triangle ABC.LinePA meets the incircle at two points; of these let X be the point that is closer to A. In a similar manner, let Y and Z be the points where PB and PC meet the incircle respectively. Prove that DX, EY, CZ are concurrent.

MG1371.912=MG1377.913,S922 Let D, E, F be points on the sides BC.CA, AB of ABC. Let U, V, W, X, Y, Z be the midpoints of BD,DC,CE,EA,AF,FB respectively. Prove that 1 UVW + XY Z − DEF 2 is a constant independent of D, E, F.

MG1372.912 For which angles θ, a rational number of degrees, is it the case that tan2 θ + tan2 2θ is rational?

MG1377.913.S923=MG1371.912 Let DEF be a variable triangle inscribed in ABC, and let U, X, V, Y, W, Z be the midpoints of the line segments BD,DC,CE,EA,AF and FB respectively. Show that 1 UVW + XY Z − DEF 2 is constant.

MG1386.915.S925.(Fukuta) Let ABC be an acute-angled triangle, let H be the foot of the altitude from A,andletD, E, Q be the feet of the perpendiculars from an arbitray point P in the triangle onto AB, AC, AH respectively. Prove that AB · AD − AC · AE = BC · PQ. YIU : Problems in Elementary Geometry 196

MG1395.922.S932.(Sadovenau) Let A1A2 ···An be an n−gon circumscribing a circle, and let B1,B2,...,B − n denote the points of tangency of thesides. Let M be a point on the circumference of the incircle. Show that n n d(M,Bσ(i)Bσ(i+1))= d(M,AiAi+1). i=1 i=1

MG1396.922.(Fukuta) Let ABC be an arbitrary triangle, let L1 and L2 be the trisection points of BC, arranged in order from B to C. Describe a method for dissecting triangle ABL1 into four parts, each of which is a triangle or a quadrilateral, so that the parts can be reassembled to form a triangle congruent to triangle AL2C.

MG1402.923.S933.(Pˆirv˘anescu) Let ABC be a given triangle, and M, N,andP be arbitrary points in the interiors of the line segments BC, CA and AB respectively. Let lines AM, BN and CP intersect the circumcircle of ABC in points Q, R and S respectively. Prove that AM BN CP + + ≥ 9. MQ NR PS Same as Crux 1430.

MG1405.924. Two circles inscribed in distinct angles of a triangle are isogonally related if the tangents from the third vertex not coinciding with the sides are symmetric with respect to the bisector of the third angle. Given three circles inscried in distinct angles of a triangle, prove that if any two of the three pairs of circles are isogonally related that so is the third pair.

MG1409.925.S935.(G.A.Heuer) Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice points in the interior. See also Putnam Competition A3, 1990.

MG1411.925.S935.(Covas) Let C1 and C2 be nonconcentric circles in the plane, and consider the set of lines that intersect C1 and C2 in equal chords. Show that these lines are all tangent to a single parabola.

MG1418.932.S942.(Hentzel and Sprague) Given three distances a, b, c, construct (using straightedge and compass and with analytic geometry) a square ABCD and a point P such that PA = a, PB = b and PC = c. See E391.397.S406.

MG1420.932.S942.(Turcu) If α, β, γ, δ areeal numbers, and n is an odd integer, cos α + cos β +cosγ +cosδ =0,andsinα+sinβ +sinγ +sinδ =0,provethatcosnα+cosnβ +cosnγ + cos nδ =0,andsinnα +sinnβ +sinnγ +sinnδ =0. YIU : Problems in Elementary Geometry 197

MG1421.932.S942.(Fukuta) If a polygon A1A2 ···An has an inscribed circle with center I and a circumcircle with center O,andCi is the circumcenter of the triangle IAiAi+1, i = 1, 2,...,n,whereAn+1 = An, prove that the Cis are concyclic.

MG1423.933.S943.(Mocanu) Two equilateral triangles, of side lengths a and b respectively, are enclosed in a unit equivlateral triangle so that they have no common interior points. Prove that a + b ≤ 1.

MG1426.933.S943.(Fukuta) Consider a circle with center at O, and a regular n−gon A1A2 ···An, contained entirely with the given circle. Let C denote the center of the n−gon. Let PiQi, i =1, 2,...,n be the chords of the given circle that are perpendicular to CAi at Ai. n 2 2 Prove that i=1(CPi + CQi ) is constant.

MG1429.934.S944.(Wee Liang Gan) Let P be a point inside the convex n−gon A1A2 ···An. 1 − 1 Prove that at least one of the angles PAiAi+1, i =1, 2,...,n,islessthanorequalto(2 n )π. (All subscripts are taken modulo n).

1431.934.S944.(Fukuta) In the given triangle ABC,letAD, AE be any cevians from A to BC.A circle drawn through A cuts AB, AC, AD, AE, or their extensions, at the points P , Q, R, S respectively. Prove that AP · AB − AR · AD BD = , AS · AE − AQ · AC EC where AP , AB, . . . , denote the lengths of the directed line segments AP , AB,....

MG1435.935.S945.(Pˆirv˘anescu) Let A1,...,An be point masses (n ≥ 3) on the spehere S(O, R)ofradiusR and cneter O,andletG be their centroid. Let M be an arbitrary point in thesphere having OG as a diameter, and let Bk be the other intersection of MAk with the sphere S(O, R). Show that n n MBk ≥ MAk. k=1 k=1

MG1439.941.S951.(C.Vanden Eynden) All lines in the sketch below have slopes 0, ] ± 1.What point do the points Pn approach ?

MG1442.941.S951.(W.O.Egerland and C.E.Hansen) Prove that two ellipses with exactly one focus in common intersect in at most two points.

MG1444.942.S952. (C.Turcu) In the following ﬁgure, ABCD is a trapezoid with AB YIU : Problems in Elementary Geometry 198 parallel to CD,andthelengthofAB is the sum of the lengths of AC and CD. E is the midpoint of BD,andF is a point on AC such that EF is parallel to CE.Provethat (a) AE and DF are perpendicular to BF; (b) C is the incenter of triangle DEF if and only if AD is perpendicular to AB; (c) EF is parallel to AD if and only if the length of AB is 3 times the length of CD.

MG1445.942.S952. (I.V.Burkov) Let r1,r2,r3 and s1,s2,s3 be orthonormal right oriented 3 triplets of vectors in R ,andk1,k2,k3 be nonzero real numbers with diﬀerent absolute values such that k1(r1 × s1)+k2(r2 × s2)+k3(r3 × s3)=0.

Prove that ri is parallel to si, i =1, 2, 3.

MG1447.942.S952. (Pˆirv˘anescu) Let M denote an arbitrary point inside or on a tetrahedron A1A2A3A4,andletBi be a point on the face Fi opposite vertex Ai, i =1, 2, 3, 4. For each i,letMi be the point where the line through M parallel to AiBi intersects Fi. Show that

4 min ≤ MMi ≤ max AiBi. 1≤i≤4 i≤i≤4 i=1

MG1452.943.S953. (J.Frohliger and A.Zeuke) Let ABC be a given triangle and θ an angle between −90◦ and 90◦.LetA, B, C be points on the perpendicular bisectors of BC, CA, AB respectively, so that BCA, CAB,and ABC all have measure θ. Show that for all but two values of θ, the lines AA, BB and CC are concurrent, provided that points A,B,C ar not equal to A, B, C respectively.

MG1455.944.S954.(Fukuta) In a hexagon A1A2A3A4A5A6 inscribed in a circle with center O,letMi, i =1, 2,...,6 be the midpoints of the sides AiAi+1,whereA7 = A1.Provethatif M1M3M5 and M2M4M6 are equilateral, A1A3A5 and A2A4A6 are also equilateral.

MG1457.944.S954. (L.Carter et al.) (a) For a point P inside a circle draw three chords through P making six 60◦ angles at P and form two regions by coloring the six “pizza slices” alternately black and white. Prove tht the region containing the center has the larger area. (b)* Prove that if ﬁve chords make ten 36◦ angles at P , then the region containing the center has the lesser area. See also in the same issue, p.267: Proof without words: Fair allocation of a pizza.

MG1460.945.S955.(D.P.Anastssiu) Let ABC be an acute triangle with altitudes AA, BB , CC .LetA1, B1, C1 be the second intersection points of lines AA , BB , CC with the YIU : Problems in Elementary Geometry 199 circumcircle of triangle ABC. Show that

2 2 2 AA1 sin 2A + BB1 sin 2B + CC1 sin 2C>24S0,

where S0 denotes the area of triangle A B C .

MG1461.945.S955. (V.Koneˇcn´y) Given the vertices V1,V2 and foci at F1, F2 of two parabolas with the same axis, construct a common tangent, if one exists, using only a compass and straightedge. Assume that the unit of length is given.

MG1469.952. (R.Izard) In EDB shown below, A and C lie on EB and ED respectively; CB and DA intersect at F .Also, EDB =6, ECA · 14 DC AB =4,and CFA+ DFB = 5 .ProvethatDEB is a right triangle.

MG1472.952. (E.G¨rel) Let Q denote an arbitrary convex quadrilateral inscribed in a ﬁxed circle, and let F(Q) be the set of inscribed convex quadrilaterals whose sides are parallel to those of Q. Prove that the quadrilaterals in F(Q) of maximum area is the one whose diagonals are perpendicular to one another.

MG1474.953.S963. (G.A.Edgar) Consider triangle ABC with sides lengths a, b, c. Sup- pose r, r, r are positive numbers satisfying

r ≤ a, r ≤ b ≤ r + r, r ≤ c ≤ r + r; 4 r ≥ 2r, r ≥ 2r, r ≤ r. 3 What is the least possible measure of angle A?

MG1483.954.S964. (A.Teodorescu-Frumosu) Let ABC be an arbitrary triangle, and let a, b, c be the lengths of the sides BC, CA, AB respectively. Let M be the midpoint of the segment BC,letα = BAM, β = CAM,andx = AMB. Show that b a cos x = . sin α sin(α − β)

MG1487.955.S965. (E.Kitchen) Given circles C and C with centers O and O,andcircles C1 and C2 tangent to C at points M1 and M2, and internally tangent to C at points N1 and N2, prove that the lines M1N1, M2N2 and OO are concurrent. YIU : Problems in Elementary Geometry 200

MG1493.961.S971. (J.Fukuta) In ABC,letL1 and L2, M1 and M2, N1 and N2 be distinct points on the sides BC, CA, AB, respectively, such that BL CL CM AM AN BN 1 = 2 = 1 = 2 = 1 = 2 < 1. L1C L2B M1A M2C N1B N2A

Let PL1L2, QM1M2, RN1N2, SM2N1, TN2L1,andUL2M1 be the equilateral triangles built outwards on the sides of the hexagon L1L2M1M2N1N2. (i) Prove that the segments PS, QT ,andRU have equal lengths, and the lines PS, QT , RU interesect at 120◦ and are concurrent. (ii) If G1, G2, G3, G4, G5, G6 are the centroids of triangles QSR, SRT, RT P, TPU, PUQ, UQS,provethatG1G2G3G4G5G6 is a regular hexagon whose centroid coincides with that of ABC.

MG1500.963. (S.Stahl) Let r be a positive real number and let A0B0C0 be equilateral. For each n ≥ 0, let An+1 and Bn+1 divide the sides AnBn and AnCn respectively in the internal ratio r :1,andsetCn+1 = An.IfP = limn→∞ AnBnCn, prove that the measures of B0PC0, C0PA0,and A0PB0 form an arithmetic progression.

MG1506.964. (WU Wei Chao) Let I and O denote the incenter and circumcenter, respectively, of ABC. Assume ABC not equilateral. Prove that AIO ≤ 90◦ if and only if 2BC ≤ AB + CA, with equality holding only simultaneously. YIU : Problems in Elementary Geometry 201

College Mathematics Journal 1976 – 1997

CMJ33.751.S761. Let P be any point on a circle. Prove that the four distances from P to the vertices of a square inscribed in the circle cannot be all rational.

CMJ47.754.S764. sin2 A +sin2 B +sin2 C =2(sinA sin B cos C +sinA cos B sin C +cosA sin B sin C).

CMJ49.754.S764. Determine the maximum value of sin A +sinB +sinC A B C . cot 2 +cot 2 +cot 2

CMJ762.p59. A simple proof of the reﬂection property of parabolas.

CMJ61.762.S773. Construct the altitude to the hypotenuse of right triangle T0.Callone of the two subtriangles T1. Construct the altitude to the hypotenuse of T1 and call one of the subtriangle T2. Continue the process so that, in general, Tn is one of the two subtriangles formed by constructing the altitude to the hypotenuse of Tn−1. In the solution to CMJ13.744, it was ∞ shown that there exist sequences Ti,i =1, 2, 3,... for which i=1 AreaTi equals the area of T0. ∞ In any one of these sequences, let hi be the altitude to Ti−1,i=1, 2,,....Provethat i=1 hi is twice the area of T0 divided by the diﬀerence between the hypotenuse and a leg of T0. ∞ k 1 CMJ63.762.S774. Find all values of k for which n=1 tan n converges. Answer: k>1.

CMJ64.762.S774. An integer sided square is inscribed in an integer-sided right triangle so that a side of the square lies on the hypotenuse. What is the smallest possible length of the side of the square? abc Solution. The length of each side of the square is ab+c2 . Is it clear that this has the same answer as CMJ390.885.S902 ? See also MG945.754.S764.

CMJ72.763.S781. In angleABC,a>b>c. Prove or disprove a − b b − c c − a c a b A B C ( + + )( + + )=1− 8sin sin sin . c a b a − b b − c c − a 2 2 2 YIU : Problems in Elementary Geometry 202

This is true.

CMJ74.764.S781. Let H be the intersection of the altitudes of acute triangle ABC.Choose B on HB and C on HC so that ABC and ACB are right angles. Prove that AB = AC.

CMJ75.764.S781. Find an integer sided triangle such that each of its angles can be trisected with straightedge and compasses. Solution. Let S be the set of all cubes of complex numbers a + bi with a2 + b2 ∈ Z.Writing c + di =(a + bi)3, we obtain a right triangle with legs c and d.

CMJ79.771.S782. In ABC,letD, E, F be points on BC, CA, AB respectively such that AF · AB = BD· BC = CE· CA = r. Prove that the ratio of the area of the triangle determined − 3 by AD, BE, CF to the area of ABC is 4 r2−r+1 .

CMJ85.772.S783. (a) 27Rr ≤ 2s2. (b) Let O be a point within ABC and d1,d2,d3 be the distances from O to the sides BC, CA, AB respectively. Prove that abc d sin A + d sin B + d sin C = . 1 2 3 4R2

CMJ773.p142. Some consequences of a property of the centroid of a triangle: For any point P , not necessarily in the plane of ABC,

PA2 + PB2 + PC2 = GA2 + GB2 + GC2 +3PG2.

CMJ773.p152. Quasi Pythagorean triples for obligue triangles.

CMJ94.773.S785. Given a, b + c and A, 0

CMJ98.774. Prove or√ disprove 3 3 3 ≥ 8 3s 4 4 4 ≥ 2 (i). a + b + c 3 . (ii). a + b + c 16 .

CMJ781.p21. Polya, Guessing and Proving.

CMJ107.781. Prove or disprove that the radius of a circle inscribed in a Pythagorean triangle is an integral multiple of the greatest common divisor of the three sides. YIU : Problems in Elementary Geometry 203

π CMJ109.781. If A = 3 ,then

4R2(sin2 A +sinB sin C) − b2 − c2 = . 2cscA − 4cotA

CMJ110.781. In ABC, AP, BQ, CR are the altitudes, AD, BE, CF the internal bisectors of the angles. Let BE,CF intersect AP at X1,X2 respectively, CF,AD intersect BQ at Y1,Y2 respectively, and AD, BE intersect CR at Z1,Z2 respectively. Prove that

IX1 · IY1 · IZ1 = IX2 · IY2 · IZ2 = X1X2 · Y1Y2 · Z1Z2.

CMJ117.783. Let E be the intersection of the diagonals of parallelogram ABCD and let P and Q be points on a circle with center E.Provethat

PA2 + PB2 + PC2 + PD2 = QA2 + QB2 + QC2 + QD2.

CMJ118.783. Triangles ABC and DEF are inscribed in the same circle. Prove that

sin A +sinB +sinC =sinD +sinE +sinF if and only if the perimeters of the triangles are equal.

CMJ120.783. Let sin A +cosB = p and cos A +sinB = q,wherep and q are not both zero and p2 + q2 ≤ 4. Express sin(A + B), cos(A + B), sin(A − B)andcos(A − B)intermsofp and q. Three of these, except cos(A + B) are very easy to ﬁnd.

CMJ125.784.S802.() = CMJ195.813.S825.() Determine all values of x ∈ (0,π)which satisfy tan x =tan2x tan 3x tan 4x. kπ Answer. 18 ,k =1, 5, 7, 11, 13, 17.

CMJ128.784.S802. π π 3π 5π 1 tan (cos +cos +cos )= . 14 14 14 14 2

CMJ130.784.S802. Prove that 1 1 1 ≥ 9 (i) a + b + c 2s . YIU : Problems in Elementary Geometry 204

2 2 2 ≥ 4s2 (ii) a + b √+ c 3 . (iii) s2 ≥ 3 3. √ (iv) a2 + b2 + c2 ≥ 4 3. 3 3 3 ≥ 8s3 (v) a + b + c 9 . Interesting editorial comments.

CMJ131.791.S803. The lengths of the sides of an isosceles triangle are integers and its area is the product of the perimeter and a prime. What are the possible values of the prime? Answer. 2, 3, 5.

CMJ134.791.S803. Let A be the surface area of a rectangular parallelepiped, V the volume 2 and d the diagonal. Prove that 2d2 ≥ A ≥ 6v 3 .

CMJ140.792.S804. Locate a point P in the interior of a triangle such that the sum of the squares of the distances from P to the sides of the triangle is a minimum.

CMJ143.793.S805. Let AD, BE, CF be the medians of ABC.ProvethatcotDAB + cot EBC +cotFCA =3(cotA +cotB +cotC).

CMJ146.794.S811. Prove that the smallest regular n-gon which can be inscribed in a given regular n-gon will have its vertices at the midpoints of the sides of the given n-gon. cf. MG1076.794.S804, CMJ203.815.

CMJ153.794.S812.

CMJ156.801.S813.

CMJ159.801.S813. 1 n 1 π arctan + arctan = . n +1 i2 + i +1 4 i=1

CMJ160.801.S813. Triangles ABC and ABC are inscribed in a circle so that c−c = a−a. Prove that sin A +sinB +sinC =sinA +sinB +sinC if and only if (a − a)3 +(b − b)3 +(c − c)3 =3(a − a)(b − b)(c − c). YIU : Problems in Elementary Geometry 205

CMJ162.802.S814. What is the minimum number of acute angle triangles into which a square can be partitioned ? Note: This appears in CM29.S1(45).

CMJ165.802.S815.

1 CMJ175.804.S821. A triangle is equilateral if and only if = 6 (hab + hbc + hca).

CMJ187.812.S824. Find the length of a side of an equilateral triangle in which the distances from its vertices to an interior point are 5, 7, 8. Long solution by H.Eves. See also CM10.p242, Bottema, On the distances of a point to the vertices of a triangle. Here is a generalization: For three positive numbers a, b, c satisfying a ≤ b ≤ c ≤ a + b, give a euclidean construction of an equilateral triangle ABC together with a point P (not necessarily inside the triangle) such that AP = a, BP = b and CP = c. Distinguish between the cases a + b = c and a + b

CMJ812.p149. M.K.Siu, From an inequality to inversion.

CMJ813.p206. A visual application of the compound angle formula in terms of areas.

CMJ193.813.S825 AE is a chord bisecting angle A of ABC. A circle that is tangential internally to the sides AB and AC is also tangential at E to the circumcircle of the triangle. Let H be the center and r0 the radius of this inner circle. If I is the incenter and r the inradius of the triangle, prove or disprove that 2 A (i) r0 = r sec 2 ; 2 A 2R sin 2 (ii) IH = A . 1+sin 2 Remark. Durell,p.275, sin2 A +sin2 B +sin2 C =2+2cosA cos B cos C. YIU : Problems in Elementary Geometry 206

CMJ195.813.S825 = CMJ125.784.S802. Find an acute angle with measure x such that

tan x =tan2x tan 3x tan 4x.

CMJ814.p271. Vector identities from quaternions.

CMJ196.814.S831. Assume that l is a line segment which is not a median and which bisects a triangle into two polygons of equal areas. Can l contain the centroid?

CMJ198.814.S831.

CMJ203.815. Find the largest and the smallest regular polygons of n sides which can be inscribed in a given regular polygon of (n +1)sides. See also CMJ146.794.S811, MG1076.794.S804.

CMJ207.821.S832. Evaluate [ n ] 2 (2k − 1)π cos2 . 2n k=1 n Answer. 4 .

CMJ821.p61. A classroom approach to Pythagorean triples.

CMJ218.823.S834. Prove or disprove that 3π 2π √ tan +4sin = 11. 11 11 This is true.

CMJ219.823.S841. Show that the product formula 2 π π π =cos · cos · cos ··· π 4 8 16

1 generates successive approximations of π with error bound En < 3·22n−4 ,wheren is the power of two occurring in the last cosine term used.

CMJ225.824.S842. Prove or disprove (a) sin2 A+sin2 B +sin2 C =2(sinA sin B cos C +sinA cos B sin C +cosA sin B sin C)True. (b) sin2 A +sin2 B +sin2 C<2(sin A sin B +sinB sin C +sinC sin A)False. YIU : Problems in Elementary Geometry 207

A B C ≥ A B C (c) 8(cos√ 2 +cos 2 +cos 2 ) 27(tan 2 +tan 2 +tan 2 ). False. 9 3 2 2 2 ≥ 2 ≥ (d) 2 (a + b + c ) (a + b + c) (sin A +sinB +sinC) 54 .True.

CMJ227.824.S842. Let ABCD be a convex quadrilateral with AC and BD intersecting at M. Assume that P and Q are points such that MP ⊥ AB and PM intersects CD at Q.Prove or disprove (i) AB2 + CD2 = AD2 + BC2 if and only if AC ⊥ BD. (ii) If AC ⊥ BD,thenABCD is cyclic if and only if DQ = QC.

CMJ825.p329. vector approach to the Euler line.

CMJ232.825.S843. A perfect triangle is deﬁned to have integral sides and area such that the perimeter equals the area. It is known that there are only ﬁve perfect triangles:

T1 =(6, 8, 10); T2 =(5, 12, 13); T3 =(9, 10, 17); T4 =(7, 15, 20); T5 =(6, 25, 29).

Notice that each of the pairs (T1,T3)and(T1,T5) has a common side. Prove that these pairs can be placed along their common sides to form a larger triangle in each case. See also CMJ845.p429.

CJM233.825.S843. (a + b − c)a(b + c − a)b(a + c − b)c ≤ aabbcc. Equality holds if and only if the triangle is equilateral.

CMJ234.825.S844. Let E be a point in the plane of parallelogram ABCD and Let K(XY Z) denote the area of XY Z. Prove or disprove that |K(ADE) − K(ABE)| = K(ACE). This is indeed true.

CMJ831.p2,832.p154. Honsberger, the Butterﬂy Problem andother delicacies from the noble art of euclidean geometry.

CMJ831.p72. The Steiner-Lehmus theorem as a challenge problem.

CMJ238.831.S844. Let a and b denote the lengths of the legs of a right triangle and c denote the length of the hypotenuse. Prove that a2(b + c)+b2(a + c) >π. abc YIU : Problems in Elementary Geometry 208

More generally, a2(b + c)+b2(a + c) C ≥ 2+csc . abc 2

CMJ831.240.S845. A circle is externally tangent to the circumcircle of ABC and also to AB at P and AC at Q. Prove that the midpoint of PQ is the center of the escribed circle opposite to A of ABC.

CMJ244.832.S851. Prove that − n1 kπ kπ (n2 − 1)(n2 − 4) csc2 cot2 = . n n 45 k=1

CMJ249.833.S852. Prove that a point D on BC of ABC√ exists so that AD is the geometric mean between BD and DC if and only if AC + AB ≤ 2BC. When does equality hold ? Equality holds if and only if AD is the internal bisector of angle BAC.

CMJ834.p301/ Construction of integral cevians. See also CM6.p98, Equal cevians, and postscript on CM6.p239.

CMJ251.833.S852. Prove that

(a4 + b4 + c4)2 > 2(a8 + b8 + c8) if and only if a2,b2,c2 are the lengths of the sides of a triangle.

CMJ253.833.S853. A B C 3(cot A +cotB +cotC) ≥ cot +cot +cot 2 2 2 with equality if and only if the triangle is equilateral.

CMJ262.835.S854. Assume that a plane π contains both a vertex of a tetrahedron and the centroid. Prove or disprove that if π separates the tetrahedron into two solids of equal volume, then π bisects an edge of the tetrahedron. True.

CMJ835.p382. Radii of the incircle and excircles of right triangles. YIU : Problems in Elementary Geometry 209

CMJ835.p436. Ellipses from a circular and spherical point of view.

2 400π CMJ264.835.S854. (2s) > 61 .

CJM841.p52. An analytic approach to the Euler line.

CJM841.p55. Integer sided triangles with one angle twice another. Solution. Easy. Take two integers m,nsatisfying m

CJM265.841.S854. Determine the extreme values of

i i i Si =sin B +sin C − sin A where A, B, C are the angles of a triangle.

CJM268.841.S855. Show that n tan nx sec jx · sec(j − 1)x = . tan x j=1

CMJ842.p140. Reﬂection property of the ellipse and the hyperbola.

CMJ273.842.S861. Let ABCD be a tetrahedron and let α, β, γ, and δ be the areas of the faces opposite vectices A, B, C, D respectively. If the edges meeting at D are mutually perpendicular, prove that δ2 = α2 + β2 + γ2. Interesting solutions. Generalization: The square of the area of one face of a tetrahedron is equal to the sum of the squares of the areas of the other three faces minus twice the sum of the products of the areas of the other faces two at a time and the cosine of the diheral angle btween them. Proof by vector algebra.

CMJ843.p252. Proving Heron’s formula tangentially. The traditonal proof is much more beautiful. Also, CMJ872.p137 Heron’s area formula. This is the standard proof!

CMJ279.843.S862. Prove or disprove: a concyclic trapezoid is isosceles if and only if the altitude of the trapezoid is the geometric mean of the bases.

CMJ844.p326. M.K.Siu On the sphere and cylinder. YIU : Problems in Elementary Geometry 210

CMJ842.p126. Grazing goat in n dimensions.

CMJ845.p.430. Return of the grazing goat in n dimensions.

CMJ285.844.S863. Consider ABC and its excircle opposite A. Secant AP RQ meets the side BC at P and the excircle at R and Q with R between P and Q. Prove or disprove that AQ AP is maximum if and only if P is on the inscribed circle of ABC.

CMJ845.p429. Right triangles with perimeter and area equal. See also CMJ232.825, where it is asserted that there are only ﬁve triangles of integer sides, with area equal to perimeter. Two of these are right triangles, namely (6,8,10) and (5,12,13). This note shows that these are the only ones that contain right angles. Conjecture: For every natural number n, there is at least one primitive Pythagorean triangle in which the area equals n times the perimeter. See MG1088.795.S811, MG1077.794.S804, where this conjecture was resolved.

CMJ286.845.S863. Prove that ABC is equilateral if and only if √ 3 3 (cot A +cotB +cotC) · sin A sin B sin C = . 2

CMJ852.p122. constructing the foci and directrices of a given ellipse.

CMJ300.852.S871. Let AD be the bisector of angle BAC of ABC.ProvethatAD·BC = AC · AB if and only if angle A is twice one of the other angles of the triangle.

CMJ302.853.S872. Prove or disprove that a necessary and suﬃcient condition for

s(s − a)(s − b)(s − c) > · is C

(a2 + b2 + c2)(a2 + b2 + c2) − 2(a2a2 + b2b2 + c2c2) ≥ 16.

CMJ304.853.S872. Let ABCD be a convex quadrilateral with consecutive sides of lengths a, b, c, d. Prove that a necessary and suﬃcient condition that a circle can be inscribed in ABCD is that a(a − b + c + d) (a + b − c + d)d = . a + c b + d YIU : Problems in Elementary Geometry 211

See also CMJ374.882.S895.

CMJ309.854.S873. Call a point P in the plane of a triangle ABC a tangency point if there exist points A,B,C for which the following six perpendicular distances are equal: B to PA and PC, A to PB and PC, C to PA and PB. Does every triangle have a tangency point? Solution. If we do not require that B and C be on the same side of PA, and similarly for A, C and PB; A, B and PC, then each point on one of the three lines connecting the midpoints of two sides of ABC is a tangency point. But, if we do require this condition, then the incenter and the three excenters of the triangle are the only tangency point.

CMJ317.861. notation not very inspiring!

CMJ862.p167. Angling for Pythagorean triples. An acute angle 2θ is Pythagorean if and only if tan θ is rational.

CMJ325.862. Let AB be the longest side of a convex quadrilateral ABCD incsrcibed in a circle O.LetOE and OF be the two radii intersecting AB which are respectively perpendicular to the diagonals DB and AC of ABCD.LetE and F be the feet of the perpendiculars to AB from E and F respectively. Prove that EF is the arithmetic mean of AD and BC.

CMJ863p.238. Three ways to maximize the area of an inscribed quadrilateral. Among all quadrilateral inscribed in a circle, the square has the largest area. (Leroy Meyer)

CMJ331.864.S883. notations too complicated inequalities involving sides and angles of a triangle.

CMJ334.864.S884. Let P1P2P3 be a right triangle with right angle at P3.ChooseP4 on P1P2 such that P3P4 ⊥ P1P2. Then choose P5 on P2P3 such that P4P5 ⊥ P2P3.Continuethis process so that Pi will be a point of Pi−3Pi−2 with Pi−1Pi ⊥ Pi−3Pi−2,i =4, 5,....Providea euclidean construction for the limit point of Pi.

CMJ865.p392. Geometry of the rational plane, not very interesting!

CMJ865.p418. A pretrigonometry proof of the reﬂection property of the ellipse.

CMJ865.339.S891. Suppose ABC is any triangle and D, E, F are arbitrary interior points of segments BC,AC,AB respectively. Let G be the intersection of AD and EF, H that of BE YIU : Problems in Elementary Geometry 212 and DF,andI that of CF and DE.Provethat FG EI DH AF BD CE · · =( · · )2. GE ID HF FB DC EA

1 1 π CMJ872.p141. arctan 2 +arctan3 = 4 without words. CM292. Fold a square piece of paper to form 4 creases that determine angles of tangents 1, 2, 3.

CMJ346.872.S892. Given ABC, determine the point P in it (or on the boundary) which maximizes the sum PA+ PB + PC. (The corresponding problem with maximum replaced by minimum is well known). The same problem also appears as Crux 2215.972.

CMJ347.872.S892. Suppose the cevians AD, BE, CF of ABC mutually trisect each AF · BD · CE other. Evaluate FB DC EA.

CMJ351.873.S893. Let a ≤ b 6abc.

CMJ353.873.S893. Prove that the interior of a triangle contains a point that lies on three congruent circles, each tangent to two diﬀerent sides of the triangle.

CMJ354.873.S893. Let ABC be an integer sided right triangle. Let CP and CQ be, respectively, the median and altitude of the hypotenuse AB. Under what condition are the sides of CPQ integers?

CJM874.p300. Equiangular lattice polygons and semiregular lattice polyhedra. This contains the theorem that a regular n-gon can be embedded in a three-dimensional lattice if and only if n =3, 4, 6. See also Klamkin and Chrestenson, Polygons imbedded in a lattice , AMM70 (1963) 447-448. But the main theorem of the present note states that of the thirteen semiregular polyhedra, only the truncated tetrahedron, the truncated octahedron and the cuboctahedron occur as lattice polyhedra. See also CMJ821.p36, Semi-regular lattice polygons by Honsberger, and also AMM923 by Beeson.

CMJ364.875.S894. Let P be a point from which exactly two normals can be drawn to the graph of y = x2. Determine the locus of P .

CMJ365.875.S894. Let A1A2A3A4 be a quadrilateral inscribed in a circle. If Pi is the incenter of AiAi+1Ai+2,i=1, 2, 3, 4,A5 = A1,A6 = A2,provethatP1P2P3P4 is a rectangle. YIU : Problems in Elementary Geometry 213

CMJ366.881.S894. 1 1 1 1 1 1 1 1 1 ≤ ( + + )2 ≤ + + ≤ . 2rR 3 a b c a2 b2 c2 4r2

CMJ374.882.S895. Determine necessary and sufficient conditions on the consecutive sides a, b, c, d of a convex quadrilateral such that one of its configurations has an incircle, a circumcircle and perpendicular diagonals. Solution. See also CMJ304.853. A necessary and sufficient condition for the quadrilateral to have an incircle is that a+c = b+d. A necessary and sufficient condition for the quadrilateral to have a circumcircle and the diagonals of the quadrilateral to be perpendicular is that a2 +c2 = b2 + d2. Together, these imply a = b, c = d or a = d, c = b. This is the desired condition.

CMJ376.883.S895. In ABC, AP, BQ, CR are the bisectors, and P ,Q,R are points on BC, CA, AB such that PP//AB, QQ//BC, RR//CA.Provethat 1 1 1 1 1 1 + + =2( + + ). PP QQ RR a b c

CMJ382.884. Let αi,i =1, 2, 3, 4 be the angles of an inscribed quadrilateral. Prove or disprove 4 α 4 α cot i < 2 csc i . 2 2 i=1 i=1

CMJ383.884. 18abc √ a2 + b2 + c2 − 2bc − 2ca − 2ab + ≥ 4 3. a + b + c

CMJ385.884. √ π 3π 5π 7 cos cos cos = . 14 14 14 8

CMJ390.885.902. Find the smallest Pythagorean triangle in which a square with integer sides can be inscribed so that a side of the square coincides with the hypotenuse of the triangle. In what sense ? See also CMJ64.762.S774, MG945.754.S764.

CMJ395.891.S903. If the altitudes of an acute triangle ABC are extended to intersect its circumcircle in points A,B,C” respectively, prove that AreaABC ≤ AreaABC. YIU : Problems in Elementary Geometry 214

CMJ892.p134. To view an ellipse in perspective. ∞ 2 CMJ399.892.S903. Evaluate n=1 arctan n2 .

CMJ893.p232. Lattices of trigonometric identities.

CMJ408.894.408.S905. Let A,B,C be the excenters of ABC. The perapendiculars from B to AB and from C to AC meet in a point A;pointsB and C are determined analogously. Prove that the lines AA,BB,CC are concurrent.

CMJ417.901.S911. Let ABC be a triangle, the lengths of whose sides are a, b, c.LetI denote the incenter and E the excenter opposite the vertex A.LetP and Q (S and R)bepoints on the sides (extensions of the sides) AB and AC such that segment PQ (SR) is parallel to side BC. (a) Prove that (i) PQ = PB+QC if and only if PQpasses through I, and (ii) SR = SB+RC if and only if SR passes through E. (b) Determine the lengths of the sides of the trapezoid PQRS in terms of a, b and c.

CMJ421.902.S912. Let P be any point on the median to side BC of ABC.Extendthe line segment BP(CP) to meet AC(AB)atD(E). If the circles inscribed in AP E and CPD have the same radius, prove that AB = AC.

CMJ903.426.S913. A circle with center O passes through vectex A of ABC and cuts sides AB and AC in points D and E respectively. AO extended meets BC at F .IfangleAF C is 60 degrees, show that AD · AB − AE · AC = AO · BC.

CJM430.903.S913. Let ABC be an isosceles triangle with AB = AC. Assume there exists points Pi, 1 ≤ i ≤ k such that (a) Pi ∈ AB if i is odd, (b) Pi ∈ AC if i is even, (c) Pi ∈ APi+2if 1 ≤ i ≤ i +2≤ k, (d) Pk−1Pk = BC (f k is even Pk = C;ifk is odd Pk = B), (e) AP1 = PiPi+1, 1 ≤ i ≤ k − 1.

CMJ443.911.S921. Let n be a positive integer. Evaluate

1 n 1 . n 2kπ k=1 3+2cos n YIU : Problems in Elementary Geometry 215

The published solution is a primitive root approach. But I think it is easier to handle with the theory of equations, starting with the Chebyshev polynomials.

CMJ444.911.S921. Given two right or obtuse triangles with sides of lengths a>b≥ c and A>B≥ C and perimeters p and P respectively; prove that aA + aP + Ap ≥ pP .

CMJ450.912.S922. A spherical ball is placed in a corner of a rectangular room so that the ball is tangent to the ceiling and to both walls. What must the ratio of the radius of that ball to the radius of another spherical ball be if the second ball is to be placed between the ﬁrst ball and the corner and is to be tangent to the ﬁrst ball as well as the ceiling and the two walls ? Generalize to n-dimensional hyperspheres. − 2 2 Solution.√ Let R and r be the radii of the spheres. n(R r) =(R + r) .Fromthis r √n−1 R = n+1 .

CMJ455.913.S923.(R.S.Tiberio) Characterize those triangles such that the angle bisectors from one vertex, the median from a second vertex, and the altitude from the third vertex are concurrent. c Solution. cos A = b+c . See also AMME263.37p104,599 concerning the euclidean construction of this triangle.

CMJ923.p106 is a very interesting reprint of Bankoﬀ on the shoemaker’s knife.Seealso CM3.p60,p.216.

CMJ923.p118, Johnson’s theorem: If three circles of equal radii intersect in a point O, their remaining intersection points lie on a circle of the same radius. Moreover, the orthocenter of these three points is O.

CMJ462.915.S925.(J.Fukuta) Through any point O inside a given equilateral triangle ABC, lines UR, QT amd SP are drawn parallel to the sides BC, CA, AB respectively, where P , Q; R, S; T , U are on the sides BC, CA and AB respectively. PRove that the area and perimeter of the hexagon PQRSTU are both minimized when O is the centroid of the triangle.

CMJ467.921.(Rabinowitz) The cosines of the angles of a triangle are in the ratio 2 : 9 : 12. Find the ratio of the sides of the triangle. See also CM10.p36, Bottema and Sauv´e. Solution. Let A, B, C be the vertices of the triangle, H its orthocenter, and X, Y, Z the projections of H on the sides BC, CA, AB respectively. If a, b, c denote the lengths of the sides of the triangle, and α, β, γ their opposite angles, it is well known that AH : BH : CH =cosα :cosβ :cosγ. YIU : Problems in Elementary Geometry 216

From this it follows that

1 1 1 : : = AX : BY : CZ a b c = AH + HX : BH + HY : CH + HZ =cosα +cosβ cos γ :cosβ +cosγ cos α :cosγ +cosα cos β. It remains to determine the cosines of the angles. Since cos α :cosβ :cosγ = 2 : 9 : 12, we may write cos α =2x, cos β =9x, cos γ =12x for some positive number x.Since 2cosα cos β cos γ +cos2 α +cos2 β +cos2 γ − 1=0, we have 432x3 + 229x2 − 1=0. 1 It is easy to check that x = 16 is the only positive root of this equation. From this, 1 9 3 cos α = , cos β = , cos γ = . 8 16 4 Consequently, from (1), a : b : c =6:5:4.

CMJ473.922.S932.(K.R.S.Sastry) A triangle ABC is called self-median if it is similar to the triangle the lengths of whose sides are the lengths of the medians in ABC.LetG be the centroid of a non-isosceles triangle ABC. Prove that the bisectors of ABC and AGC intersect on the side AC if and only if ABC is self - median.

CMJ477.923.S933.(N.Schaumberger) Let p denote the perimeter of ABC.Provethat √ a sin A + b sin B + c sin C 3 ≥ sin A sin B sin C. p

CMJ480.923.S933.(I.Sadoveanu) Let A,B,C denote the feet of the altitudes in the triangle ABC, lying on the lines BC, CA, AB respectively. Show that AC = BA = CB if and only if ABC is an equilateral triangle.

CMJ482.924.S934.(I.Sadoveanu) Let A,B,C denote the points of tangency of the sides of triangle ABC to its incircle, opposite A, B and C respectively. Let M be any point on the incircle. Prove that d(M,AB) · d(M,BC) · d(M,CA)=d(M,AB) · d(M,BC) · d(M,CA). YIU : Problems in Elementary Geometry 217

CMJ486.925.S935.(J.Sarkar) Let X, Y, Z be points on sides BC, CA, AB respectively of ABC such that AZ : ZB = BX : XC = CY : YA. Suppose that AX intersects BY at P , BY intersects CZ at Q and that CZ intersects AX at R.IfP , Q, R do not coincide, prove that

AR : RP : PX = BP : PQ : QY = CQ : QR : RZ.

CMJ490.925.S935.(J.Fukuta) Let X be a ﬁxed point within a given circle. Let A, B, C and D be variable points on the circle such that AC and BD are perpendicular chords through X.ForeachX, ﬁnd the maximum and minimum of (a) the area of the quadrilateral ABCD, and (b) the sum of the lengths of AC and BD.

CMJ493.931.S941.(K.R.Sastry) Let ABCDE be a convex, aﬃne regular pentagon in which each side is parallel to a diagonal. Let P, Q, R, S, T be points on the sides CD, DE, EA, AB, BC respectively, so that AP , BQ, CR, DS, ET concur at X. Describe the set of points X for which XP XQ XR XS XT + + + + PA QB RC SD TE is constant.

CMJ501.933.S943.(K.R.S.Sastry) Let ABC be an integer-sided right triangle, with C = 90◦. The gcd of the lengths of its sides is one. Prove or disprove: its semiperimeter is a triangular if and only if either a and c or b and c are consecutive integers. See also CMJ535.944.

CMJ503.933.S943.(L.R.Bragg and J.W.Grossman) How many triangles are there with integral sides of length at most n ? See also Crux 19.

CMJ507.934.S944.(J.Fukuta) Let a, b, c, d be the lengths of the successive sides of a quadrilateral, and let a be the maximum length. (a) Prove that if b ≥ d,orifb ≤ d and c ≤ d,

bc cd da ab + + + ≥ a + b + c + d. a b c d (b) Show that the inequality may fail without the given restrictions on b, c and d.

CMJ508.934.S944.(K.R.S.Sastry) Find all Pythagorean triangles whose sides are, respectively, n−,(n +1)− and (n +2)−gonal numbers all of the same rank. YIU : Problems in Elementary Geometry 218

CMJ511.935.S945.(Zhang Zaiming) Let ai,bi,ci, i =1, 2 be the length of the sides of triangles with area i.Provethat 2 2 2 2 2 2 ≥ a1a2 + b1b2 + c1c2 16 1 2, where equality holds if and only if the two triangles are equilateral.

CMJ515.935.S945.(L.Hoehn) Let ABCD be a quadrilateral with AB = b, BC = c, CD = d,andDA = a.ProvethatABCD is a parallelogram if and only if

ab cos A + bc cos B + cd cos C + da cos D =0.

CMJ517.941.S951.(K.S.Sadati) Let a, b, and c (a

CMJ529.943.S952. (J.Fukuta) Let A, B, C,andD be consecutive vertices of a rectangle. Find the locus of points P interior to ABCD such that PA· PC + PB · PD = AB · BC.

CMJ531.944.S954. (Klamkin and A.Liu) If A, B, C and D are consecutive vertices of a quadrilateral such that DAC =55◦ = CAB, ACD =15◦,and BCA =20◦, determine ADB.

CMJ535.944.S954. (H.Sedinger) Prove that thee exist an inﬁnite number of right triangles with sides of integer lengths a, b, c such that b>a, c>b+1,thegcdofa, b, c is 1, and the semiperimeter is a triangular number. See also CM501.933.S943.

CMJ537.945.S955. (H.G¨ulicher) In P1P2P3 a cevian through P1 cuts side P2P3 in point Q1.LetK ∈ P1Q1,(K = P1), S2 ∈ P1P3,andR3 ∈ P1P2 be such that S2K//P1P2,and R3K//P1P3.ProvethatP1Q1, S2K,andR3K are concurrent if and only if |P S | |P P | |P Q | 1 2 · 1 2 · 3 1 =1. |P1P3| |P1R3| |P2Q1|

CMJ538.945.S955. (M.S.Klamkin) Determine the maximum area of the quadrilateral with consecutive vertices A, B, C,andD if A = α, BC = b and CD = c are given.

CMJ539.945.S955. (N.Juri´c) In how many ways can three vertices of an n−dimensional cube be chosen so that the chosen vertices form an equilateral triangle ? YIU : Problems in Elementary Geometry 219

CMJ541.951.S961. (V.Oxman) Let E be an ellipse in the plane with foci F1 and F2;and let L denote a line in the plane that does not intersect E.LetA be any point on L (except the point of intersection of L with the line through F1 and F2). Construct, using only an unmarked straightedge, a point B on L and a point C on E so that an ellipse with foci A and B is tangent to E at C.

CMJ543.951.S961. (K.R.S.Sastry) A parallelogram is called self-diagonal if its diagonals are proportional to the sides. Let ABCD be a parallelogram in which AB > BC,angleA is acute, E is the midpoint ofAB,andF is chosen so that CEDF is also a parallelogram. Prove that ABCD and CEDF are congruent if and only if both are self-diagonal.

CMJ545.951.S961. (J.Fukuta) Let a, b, c, d be positive real numbers. (a) Prove that a + b>|c − d| and c + d>|a − b| are necessary and suﬃcient conditions for there to exist a convex quadrilateral that admits a circumcircle and whose side lengths, in cyclic order, are a, b, c, d. (b) Find the radius of the circumcircle.

CMJ546.952.S962. (K.R.S.Sastry) In ABC,lettheanglesatB and C be acute. Sup- pose the altitudes from A intersects BC at D,andletE and F denote the points of intersection of AD with the bisectors of the angles at B and C respectively. Prove that if BE = CF,then triangle ABC is isosceles. AD 1 1−2s2 B 1−2t2 Solution. BE = · B = AD· − 2 ,wheres := sin . Similarly, CF = AD· − 2 , tan B cos 2 2s(1 s ) 2 2t(1 t ) C with t := sin 2 .IfBE = CF,then 2s(1 − s2)(1 − 2t2)=2t(1 − t2)(1 − 2s2), s − s3 − 2st2 +2s3t2 = t − t3 − 2s2t +2s2t3 (s − t) − (s3 − t3)+2st(s − t)+2s2t2(s − t)=0 (s − t)[1 − (s2 + st + t2)+2st +2s2t2]=0 (s − t)[1 − s2 − t2 + st +2s2t2]=0 (s − t)[(1 − s2)(1 − t2)+st(1 + st)] = 0.

Since B and C are acute, 0

CMJ548.952.S962. (M.Golomb) Suppose P is a parallelogram with sides a ≥ b and obtuse angle θ. ≥ b (i) Prove that there exists a square Q in which P is inscribed if and only if sin θ +cosθ a . (ii) Prove that Q is unique if and only if P is not a square.

CMJ551.953.S963. (K.R.S.Sastry) In a triangle with integral sides and integral area (a YIU : Problems in Elementary Geometry 220

Heronian triangle) prove that a median and a side cannot be of the same length.

CMJ553.953.S963. (M.Golomb) Given a regular n−gon with center C, vertices A1, A2, ..., An and a line L in the plane of the polygon, let pk be the length of the projection of CAk n 2m onto L. Show that for each integer m,0< 2m

CMJ558.954.S964. (J.Fukuta) Prove that for any triangle ABC, there exists one and only one set of points D, E, F satisfying: (a) D lies on side BC, E lies on side CA,andF lies on side AB; (b) EA + AF = BC; FB + BD = CA; DC + CE = AB;and (c) AD, BE,andCF are concurrent. Solution. Denote by a, b, c the lengths of the sides of the triangle: BC = a, CA = b,and 1 AB = c,andbys = 2 (a + b + c) the semi-perimeter. FirstnotethatifD, E,andF are respectively the points of contact of the sides BC, CA, and AB with the excircles of the triangle on the opposite sides of A, B,andC,itiseasyto establish EA = BD = s − c; FB = CE = s − a; DC = AF = s − b. Condition (b) is clearly satisﬁed. Also, AD, BE, CF are concurrent by Ceva’s theorem.

AF BD CE s − b s − c s − a · · = · · =1. FB DC EA s − a s − b s − c The intersection of AD, BE and CF is usually called the Nagel point of the triangle. Now, we show that this is the only set of points satisfying the conditions (a), (b), (c). Let D, E,andF be a set of points satisfying the same conditions. Suppose AF = AF + J for some J.Then

EA = s − c − J, BD = s − c + J; F B = s − a − J, CE = s − a + J; DC = s − b − J, AF = s − b + J.

If AD, BE,andCF are to be concurrent, then Ceva’s theorem requires

AF BD CE · · =1. F B DC EA This means

(s − a + J)(s − b + J)(s − c + J)=(s − a − J)(s − b − J)(s − c − J), J[J2 +(s − a)(s − b)+(s − b)(s − c)+(s − c)(s − a)] = 0. YIU : Problems in Elementary Geometry 221

Since (s − a)(s − b)+(s − b)(s − c)+(s − c)(s − a) > 0. This requires J = 0. This means the points D, E,andF coincide respectively with the points D, E,andF .

CMJ564.955.S965. (C.A.Minh) Let ABCD be a convex quadrilateral. Show that if tan A +tanB +tanC +tanD =0,thenABCD is either a cyclic quadrilateral or a trapezoidal.

CMJ565.955.S965. (K.R.S.Sastry) Find the dimensions of all rectangular boxes with sides of integral lengths such that the volume is numerically equal to the sum of the lengths of the edges plus the surface area.

CMJ570.961.S971. (M.S.Klamkin) In ABC the angle bisectors of angles B and C meet the altitude AD at points E and F , respectively. IF BE = CF,provethatABC is isosceles.

CMJ574.962. (R.Patenaude) Describe the locus of the foci of all ellipses inscribed within a given nonsquare rectangle, i.e., tangent to all four sides of the rectangle.

CMJ577.963.S973. (K.R.S.Sastry) A convex heptagon A1A2A3A4A5A6A7 is such that ◦ the angle at A1 is 90 and AiAj//AkAm if i + j ≡ k + m (mod 7). Prove that

2 sin 2A2 A5A6 = 2 . sin 2A5 A2A3

CMJ578.963.S973. (R.Patenaude) Determine the lengths of the sides of a triangle with the properties that (i) the sides have integral length and (ii) one angle is twice as large as another.

CMJ583.964. (M.S.Klamkin) A known property of a parabola is that if tangents are drawn at any two points P and Q of the curve, then the line from the point of intersection of the tangents and parallel to the axis of the parabola bisects the chord PQ.Doesthisproperty characterize the parabola? That is, if a curve has the above property where the line is drawn parallel

CMJ585.965. (K.R.S.Sastry) The sides of triangle ABC are relatively prime natural numbers. The internal angle bisector of the angle at A meets BC at D.IfBD = AC,prove that (i) AB is a square, and (ii) AD is not an integer. Solution. There is only√ one such triangle ABC,withBC =3,AC =2andAB =4.The bisector AD has length 6, and BD =2,CD =1. To justify this, we denote the lengths of BC, AC,andAB by relatively prime natural numbers a, b, c respectively. Since BD = b,wehaveCD = a − b. Now, the bisector AD divides YIU : Problems in Elementary Geometry 222

BC in the ratio BD : CD = AB : AC.Fromthis,b2 =(a − b)c. Suppose a and b have a common prime divisor p,withph, pk the highest powers dividing a and b respectively. If h = k,thenpmin(h,k) is the highest power of p dividing a − b.Since − 2k min(h, k) > 0, p also divides c, contrary to the assumption that a, b,andc are relatively 2 − 2 b prime. It follows that a b =gcd(a, b) .Fromthis,c = gcd(a,b) . a − b must be divisible by p2k. We claim that a − b =1.Ifnot,everyprime divisor of a − b must divide b, and would be a common divisor of a and b. This is contrary to the assumption that a, b (and c) are relatively prime. It follows that c = b2 is a square. Indeed, a = b + 1. By the triangle inequality, b2 = c

CMJ586.965. (K.R.S.Sastry) The sides BC, CA, AB of triangle ABC are extended to the points R, P , Q, respectively, so that CR = AP = BQ.ProvethatifPQR is equilateral, then so is ABC.

CMJ595.971.S981. (J.B.Romero M´arquez) In a right triangle whose sides are a, b,and c (with a ≤ b

The angle bisectors θa and θb are given by a 2 b 2 θ2 = bc 1 − and θ2 = ca 1 − . a b + c b c + a Here,

θ2 − θ2 c(a + b + c)(a2b + ab2 +3abc + ac2 + bc2 + c3) b a = , a − b (a + c)2(b + c)2 θ2 − θ2 c(c +2a)(c3 +2ac2 +3a2c +2a3) lim b a = b→a a − b (c + a)4 c(c +2a)(c2 + ac +2a2) = (c + a)3 ar(r +2)(r2 + r +2) = . (r +1)3

Since a 2 1 2 2ar √ lim(θb + θa)=2 ca 1 − =2a r 1 − = r +2, b→a a + c r +1 r +1 we have θ − θ r2 + r +2√ lim b a = r +2. (6) b→a a − b 2(r +1)2 Combining (1) and (2), we have

m − m 3(r +1)2 1 lim b a = . 2 2 b→a θb − θa r + r +2 (r + 2)(2r +1) √ c For a right triangle r = a = 2, we have

m − m 3 lim b a = √ ≈ 0.67082 ..., b→a a − b 2 5 θ − θ 23 lim b a = 17 − √ ≈ 0.858221 ... b→a a − b 2 m − m 3 1 √ lim b a = (34 + 23 2) ≈ 0.78164 .... b→a θb − θa 14 5

CMJ 599.972.S982. (Juan-Bosco Romero M´arquez) Let ABC be a triangle with acute angles at B and C.LetH be the foot of the perpendicular from A to BC;letD be the foot YIU : Problems in Elementary Geometry 224 of the perpendiuclar from H to AB;letE be the foot of the perpendiuclar from H to AC;let P be the foot of the perpendiuclar from D to BH;andletE be the foot of the perpendiuclar from E to HC. Prove that the angle t A is a right angle if and only if AH = DP + EQ.

CMJ 604.973.S983. (J.Fukuta) The quadrilateral PQRS is inscribed in a convex quadrilateral ABCD such that P , Q, R, S are on the sides AB, BC, CD, DA respectively. Let U and V be arbitrary points on diagonals AC, BD respectively, and let A, B, C,andD be the points of intersection of UB and PQ, VC and QR, UD and RS,andVAand SP respectively. Prove that the value of the expression PA · QB · RC · SD AQ BR CS DP where now PA denotes the length of the segment PA etc), is independent of the location of points U and V .

CMJ 613.975.S985. (M.S.McClendon) Let T be an isosceles triangle with congruent sides of length k and inradius 1. Suppose that these conditions determine T uniquely. Find the length of the altitude on the third side of T .

CMJ 624.982.S992. (Harry Sedinger) LetC be a point on the line segment AB with |AC| = a>0and|CB| = b>0. Let r be a ray beginning at B and making an angle θ with AB,0<θ<π. Show that there is a point P on r that maximizes angle AP C,andshowhtat hte distance |BP| is independent of the choice of θ.

CMJ 629.983.S993. (D.Beran) In triangle ABC, the angle bisectors of angles B and C meet the median AD at points E and F respectively. If BE = CF, prove that triangle ABC is isosceles.

CMJ 635.984.S994. (S. Zimmermann) Consider a circle of radius r and an interior point P that is p units from the center of the circle. (a) Show that, for any pair of chords of the circle that intersect P at right angles, the sum of the squares of their lengths is always the same. (b) Find this sum as a function of r and p.

CMJ 650.992. (K.Korbin) Suppose that ABC is an equilateral triangle with side s and that D is a point between B and C.Letr1 and r2 dentoe the radii of the circles inscribed in triangles ABD and ACD respectively. Express s as an explicit function of r1 and r2.

CMJ 664.995. (J. Fukuta) In a triangle ABC with incenter I,leta, b, c be the lengths of the sides BC, CA, AB respectively. Let [O, r] denote the circle with center O and radius r.Let YIU : Problems in Elementary Geometry 225

U be the radical center of [A, a], [B,b]and[C, c], and let V be the radical center of [A, b + c], [B,c + a]and[C, a + b]. Prove that the points I, U, V are collinear and that I is the midpoint of UV. This is the same as AMM 10734.994 (van Lamoen). YIU : Problems in Elementary Geometry 226

Geometry Problems in Pi Mu Epsilon Journal 1949 – 1998

PME1.49F.S51S.(Leo Moser) Prove the following construction for ﬁnding the radius of a circumference. With any point O on the circumference as center and any convenient radius describe an arc PQR, cutting the given circumference in P and Q.WithQ as center and the same radius describe an arc OR cutting PQR in R, R being inside the circumference. Join P and R, cutting the given circumference in L.ThenLR is the radius of the circumference. (This is known as Swale’s construction, and is probably the simplest solution of the problem yet discovered.)

PME3.49F.S50S.() The lengths of the sides of a triangle are the roots of the cubic equation ax3 + bx2 + cx + d = 0. Find the area of the triangle. − b − − b Solution. The semiperimeter is s = 2a . Putting x = s y = (y + 2a ), we have b b b −a(y + )3 + b(y + )2 − c(y + )+d =0; 2a 2a 2a −(2ay + b)3 +2b(2ay + b)2 − 4ac(2ay + b)+8a2d =0; 8a3y3 +4a2by2 − 2a(b2 − 4ac)y − (b3 − 4abc +8a2d)=0.

If x1,x2,x3 are the sides of the triangle, the roots of this equation are s − x1, s − x2 and s − x3. It follows that b3 − 4abc +8a2d s(s − x )(s − x )(s − x )= 1 2 3 8a3 and 1 1 = −b(b3 − 4abc +8ad)= b(4abc − b3 − 8a2d). 4a2 4a2 Note that if we assume a>0, then b and d are negative, and c is positive.

PME4.49F.S50S.(Leo Moser) Towards the bottom of p.268 of Cajori’s A history of Math- ematics (1926) we ﬁnd the following statements: “Nepoleon proposed to the French mathemati- cians the problem, to divide the circumference of a circle into four equal parts by the compasses only. Mascheroni does this by applying the radius 3 times to the circumference; he obtains the arcs AB, BC, CD;thenAD is a diameter; the rest is obvious. Show how the “obvious” part of the problem may be accomplished. YIU : Problems in Elementary Geometry 227

Solution. (John A. Dyer, University of Alabama) Napoleon’s problem may be stated as follows: By use of compass only ﬁnd the side of a square inscribed in a given circle O,radius R, given also the consecutive points A, B, C, D on the circumference such that AB = BC = CD = R,andAD is a diameter. Now, with A as center and radius AC,drawcircleA. Similarly, with D as center and radius DB,drawcircleD. Circles A and D intersect at points, say E and E.ThenOE is the required side of the inscribed square.

Comment. It is not clear how the vertices of the square can be marked on the given circle. Here is a slight variation that works. Given O(A). Construct A(O) to intersect O(A)atB; B(O)to intersect A(O)atC; C(A) to intersect A(O)atD. Here, OD is a diameter of A(O). Let E be an intersection of the circles O(C)andD(B). Construct A(E) to intersect the circle O(A)at P , Q. A, P , Q are three vertices of the square. The fourth one is easy to determine.

PME6.49F.S52S,52F,55F.(Trigg) Starting with a straight edge, closed compasses, and two straight line segments a and b, construct the harmonic mean of a and b in the least number of operations. Changing the opening of the compasses, drawing a circle or the arc of a circle, and drawing a straight line are each considered an operation. The first solution, by Trigg, has 10 operations. The second solution, by Bankoff, has 9 operations. The third solution, also by Bankoff, has 8 operations.

PME8.49F.S55F.(R.T.Hood) Consider the stereographic projection of a sphere onto a plane tangent to it at its south pole S, the center of projection being the northpole N.Prove that every great circle on the sphere not passing through N is mapping into a circle whose center is on the line through N which is perpendicular to the plane of the great circle.

PME9.49F.S55F.() If the bases of a prismatoid are equal in area, then so are the sections equidistant from the midsection.

PME14.50S.S53S.(Trigg) (a) How may a sealed envelope be folded into a rectangular parallelipiped if overlapping is permitted ? (b) What is the maximum volume so obtainable in terms of the edges a and b of the envelope ? (c) What must be the relative dimensions of the envelope in order to yield a cube ? (d) What will be the volume of the cube ?

PME16.50S.S51F.(W.J.Jenkins) Given a circle and two exterior points not in a straight line with the center. Construct a circle passing through these two points and dividing the given circle into two equal arcs. YIU : Problems in Elementary Geometry 228

PME18.50S.S51F.(L.J.Burton) Points A1,B1,C1 are chosen on the sides BC, CA, AB 1 1 1 of triangle ABC such that AC1 = 2 C1B, BA1 = 2 A1C, CB1 = 2 B1A. The lines AA1, BB1, CC1 determine a triangle A2B2C2. Show that the area of A2B2C2 is one seventh of the area of ABC.

PME23.50F.S51S.(R.Dubisch) If in a triangle with sides a, b and c,wehavec ≥ b, c ≥ a, ﬁnd k such that c2 = ka2 + b2.

2 PME24.50F.S51S.(P.J.Schillo) If θn is the angle opposite the side of length 4n in the integer right triangle with sides 4n2, 4n4 − 1and4n4 +1,wheren is any positive integer, show that ∞ lim θi n→∞ i=1 is a right angle.

PME34.51F.S52S.(J.S.Frame) For what values of k are the following twelve points (0, ±k, ±1), (±1, 0, ±k), (±k, ±1, 0), the vertices of a regular icosahedron ?

AM p PME38.52S.S52F(Trigg) In the triangle ABC, AD is a median. Prove that if MD = q , p then CM extended divides AB in the ratio 2q .

PME42.52S.S52F.(M.Stover) Prove that the volume of a tetrahedron determined by two line segments lying on two skew lines is unaltered by sliding the segments along their lines (but leaving their lengths unaltered).

PME43.52S.S52F.(P.W.Gilbert) Four solid spheres lie on top of a table. Each sphere is tangent to the other three. If three of the spheres have the same radius R, what is the radius of the fourth sphere ?

PME51.52F.S53F.(Trigg) Suppose D is the foot of the altitude from C,thevertexofthe right angle in the triangle ABC. Show that the area of the triangle determined by the incenters (a+b−c)3 of triangles ABC, ADC and BDC is 8c .

PME54.S53S.S53F.(F.L.Miksa) Given a right triangle ABC, with right angle at C, ﬁnd apointP on AC so that the inscribed circles of the triangles BPC and BAP will be equal.

PME63.54S.S54F.(Bankoﬀ) State and solve the problem suggest by the following diagram: OA and OB are two perpendicular radii of a quadrant of a circle. Two circles are drawn inside YIU : Problems in Elementary Geometry 229 theincircleofOAB, each touching each other and the incircle internally. Another circle is drawn tangent to the quadrant and touching the incircle of OAB with AB as a common tangent.

PME65.54S.S69F.(M.Schechter) Prove that every simple polygon which is not a triangle has at least oen of its diagonals lying entirely inside of it. See also PME335.74F.

PME66.54S.S54F.(Trigg) If three circles with radii a, b,c are externally tangent, there are two circles with radii r, R which touch the three circles. Show that 1 1 1 1 1 − =2( + + ), r R a b c and that 1 1 a + b + c + =4 . r R abc

PME68.54F.S55F.(Bankoﬀ) An ellipse of maximum area is inscribed in a given triangle. Show that the area of the smallest quadrilateral circumscribing this ellipse is less than the geometric mean and greater than the harmonic mean of the areas of the ellipse and the triangle.

PME73.55S.S69F.(V.Th´ebault) Construct three circles with given centers such that the sum of the powers of the center of each circle with respect to the other two is the sum.

PME74.55S.S56S.(H.Helfenstein) Prove that every convex planar region of area π contains two points two units apart.

PME75.55S.S58S.(Bankoﬀ) A line parallel to hypotenuse AB of a right triangle ABC passes through the incenter I. The segments included between I and the sides AC and BC are designated by m and n. Show that the area of the triangle is given by √ √ mn(m + m2 + n2)(n + m2 + n2) . m2 + n2

PME79.55F.S56F.(Trigg) Find the bounding values of the ratio the sides a and c pfa triangle in order that the median to one side and the symmedian to the other side may be concurrent with the internal bisector of the included angle.

PME80.55F.S57S.(H.Helfenstein) Prove that the circumscribing circles of four triangles determined by four planar lines of general position have a common point. YIU : Problems in Elementary Geometry 230

PME87.56S.S57S.(E.P.Starke) The centroid G of triangle ABC is actually the center of area of ABC. Determine K, the centroid of the triangle considered as being composed of three linear segment. Show how to construct K and ﬁnd some interesting geometric properties of this point.

PME92.56F.S58S.(Bankoﬀ) It has been said that algebra is but written geometry and geometry is but diagramatic algebra. (Sophie Germain, Memoire sur les surfaces elastiques). In the spirit of this quotation, show geometrically that

∞ − n 1 =2. 2 n=2

PME96.57S.S58S.(Bankoﬀ) Acircle(p) touches the diameter AB of a semicircle (O)in D,andarcAB ofthe semicircle in R,(AD < DB). The perpendicular to OR at P cuts the arc 2 2 − 2 AD RB in S.IfRS = DB AD , ﬁnd the ratio DB .

1 PME97.58S.S58F.(A.J.Goldman) Prove that a triangle of area π has a perimeter greater than 2.

PME100.58S.S58F. (Bankoﬀ) A right triangle ABC (AC > CB is inscribed in a semicircle O whose diameter is AB. The radius OS, perpendicular to AB,cutsAC in R,andCD is the SO altitude upon AB.FindtheratioRO for which triangles ODC and CDB are both Pythagorean.

PME102.58F.S69S.(L.Moser) Give a complete proof that two equilateral triangles of edge 1 cannot be placed, without overlap, in the interior of a square of edge 1.

PME106.58F.S61F,62F.(Klamkin) An equi-angular point of an oval is deﬁned to be a point such that all intersecting chords through the point form equal angleswith the oval at both points of intersection (on the same side of thechord). It is a known elementary theorem that if all interior points of an oval are equi-angular, then the oval is a circle. (a) Show that if one boundary point of an oval is equi-angular, the oval is a circle. (b) Determine a class of non-circular ovals containing at least one equi-angular point. (c) It is conjectured that a non-circular oval can have, at most one equi-angular point.

PME116.59F.S63S.(Klamkin) Problem 147, due to Auerbach - Mazur, in the Scottisch book of Problems is to show that if a billiard ball is hit from one corner of a billiard table having commensurable sides at an angle of 45◦ with the table, then it will hit another corner. m Consider the more general problem of a table of dimension ration n and initial direction of ball YIU : Problems in Elementary Geometry 231

−1 1 of θ =tan b ,(m,n,a,bare integers). Show that the ball will ﬁrst strike another corner after an+bm − (an,bm) 2 cushions. Furthermore, determine which other corner the ball will strike.

PME120.60S.(M.Goldberg) (a) All the orthogonal projections of a surface of constant width have the same perimeter. Does any other surface have this property ? (b) A sphere may be turned through all orientations while remaining tangent to the three lateral faces of a regular triangular prism. Does any other surface thave this property ?

PME121.60S.S63S.(Klamkin) Three circular arcs of ﬁxed total length are constructed, each passing through two diﬀerent vertices of a given triangle, so that they enclose the maximum area. Show that the three radii are equal.

PME124.60F(H.Kaye),61S(Klamkin,correction),S62F. Prove the impossibility of constructing the center of a circle with a straightedge only, given a chord and its midpoint. Construct the center of an ellipse with a straightedge only, given a chord and its midpoint.

PME129.61S.S61F.(L.Moser) If R be a regular polyhedron and P a variable point inside or on R, show that the sum of the perpendicular distances to the faces of R, extended if necessary, is a constant.

PME130.61S.S62S.(H.Kaye) If P is a variable point on the circular arc AB, show that AP + PB is a maximum when P is the mid-point of the arc AB.

PME135.61F.S62F.(T.E.Hull) Suppose that k points are placed uniformly around the circumference of a circle with unit radius. Show that the product of the distances from any one point to the others is equal to k, for any k>1.

PME136.61F.(M.Goldberg) What is the smallest convex area which can be rotated continuously within a regular pentagon while keeping contact with all the sides of the pentagon ?

1 1 1 PME137.61F.S63F.(L.Moser) Show that the squares of sides 2 , 3 ,..., n ,...canallbe placed without overlap inside a unit square.

PME140.62S.S63F.(M.Goldberg) What is the smallest area within which an equilateral triangle can be turned continuously through all orientations in the plane. Editor’s remark: This problem is unsolved, and similar unsolved ones exist for the square and other regular polygons. YIU : Problems in Elementary Geometry 232

PME141.62S.S63F.(D.J.Newman) Determine conditions on the sides a and b of a rectangle in order that it can be imbedded in a unit square.

PME148.62F.(Klamkin) If a convex polygon has three angles of 60◦, show that it must be an equilateral triangle. Solution. If an n − gon, n>3, has three 60◦ angles, the remaining n − 3 angles add up to 2n − 4 − 2=2(n − 3) right angles. These angles cannot be all equal, for otherwise, each of them would be a straight angle. Now, it is easy to see that at least one of these angles exceeds 2 right angles, contradicting the convexity of the polygon. Thus, n = 3, and this is an equilateral triangle.

PME153.63S.S64F.(Klamkin) Show that the maximum area ellipse which can be inscribed in an equilateral triangle is the inscribed circle.

PME156.63F.S65S.(K.S.Murray) If A and B are ﬁxed points on a given circle and XY is a variable diameter, ﬁnd the locus of point P .

PME161.64S.S67S.(P.Schillo) It is conjectured that the smallest triangle in area which can cover any given convex polygon has an area at most twice the area of the polygon.

PME162.64S.S65F.(Klamkin) If a surface is one of revolution about two axes, show that it must be spherical.

PME165.64F.S66S.(D.J.Newman) Express cos θ as a rational function of sin3 θ and cos3 θ.

PME166.64F.S70S.(L.Moser) Show that 5 points in the interior of a 2×1 rectangle always YIU : Problems in Elementary Geometry 233 determineatleastonedistancelessthansec15◦.

PME167.64F.S66S.(Klamkin) Given a centrosymmetric strictly convex ﬁgure and an intersecting translation of it, show that there is only one common chord and that this chord is mutually bisected by the segment joining the centers.

PME169.65S.S66S.(J.Konhauser) From an arbitrary point P (not a vertex) of an ellipse lines are drawn through the foci intersecting the ellipse in points Q and R. Prove that the line joining P to the point of intersection of thetangents to the ellipse at Q and R is the nomral to the ellipse at P .

PME170.65S.S66S.(C.S.Venkataraman) Prove that a triangle ABC is isosceles or right- angled if a3 cos A + b3 cos B = abc.

PME172.65F.S66F.(J.Baudhuin) Given: semicircle O with diameter AB and equilateral triangle PAB; C and D are trisection points of the chord semicircle AB. Prove: E and F are trisection points of the chord AB. Note: A synthetic proof is desired.

PME174.65F.S66F.(C.S.Venkataraman) Find the locus of a point which moves such that the squares of the lengths of the tangents from it to three coplanar circles are in arithmetic progression.

R 2s2 PME177.66S.S67S.(C.S.Venkataraman) 2 ≥ r r1r2r3 .

PME178.66S.S67S.(K.S.Murray) Show that the centroid of ABC coincides with the centroid of ABC,whereA, B, C are the midpoints of BC, CA and AB respectively. Generalize to higher dimensions.

PME180.66S.S67S.(R.C.Gebhart) In the ﬁgure, AB = AC and ABC =90◦.The arcs YIU : Problems in Elementary Geometry 234 are both circular with the inner one being tangent to AB at A and BC at C. Determine the area of the cresent.

PME181.66S.S67F.(D.W.Crowe and Klamkin) Determine a convex curve circumscribing a given triangle such that (1) the area of the four regions (3 segments and a triangle) formed are equal, and (2) the curve has a minimum perimeter.

PME187.67S.S68S.(R.C.Gebhart) A semicircle ACB is constructed on a chord AB of a unit circle. Determine the chord AB such that the distance from O to C is a maximum.

PME189.67S.S68S.(Bankoﬀ) If A, B, C, D, E, F and G denote the consecutive vertices of a regular heptagon, show that CD is equal to the harmonic means of AC and AD.

PME191.67S.(Rabinowitz) Let P and P denote points inside rectangles ABCD and ABCD respectively. If PA = a + b, PB = a + c, PC = c + d, PD = b + d, P A = ab, P B = ac, P C = cd,provethatP D = bd.

PME195.67F.S68F.(Bankoﬀ) Mathematics Magazine (January 1963), p.60, contains a short paper by Dov Avishdom, who asserts without proof that in the adjoining diagram, AN = NC + CB.Giveaproof. YIU : Problems in Elementary Geometry 235

PME198.67S.S68F.(Rabinowitz) A semi-regular solid is obtained by slicingoﬀ sections from the corners of a cube. It is a solid with 36 congruent edges, 24 vertices and 14 faces, 6 of which are regular octagons and 8 are equilateral triangles. If the length of an edge of this polytope is e,whatisitsvolume? See also PME353.75F.

PME202.68S.S69S.(Bankoﬀ) In a right triangle, ﬁnd angle HIO given that HIO is isosceles.

PME203.68S.(Rabinowitz) Let P denote any point on the median AD of ABC.IfBP meets AC at E and CP meets AB at F ,provethatAB = AC if and only if BE = CF.

PME205.68F.S69F.(C.S.Venkataraman) ABC and PQR are two equilateral triangles with a common circumcenter but diﬀerent circumcircles. PQR and ABC are in opposite senses. Prove that AP , BQ and CR are concurrent.

PME210.68F.S69F.(Bankoﬀ) Three equal circles are inscribed in a semicircle as shown in the adjoining diagram. How is this ﬁgure related to one of the better known properties of the sequence of Fibonacci numbers ?

PME211.68F.S69F.(L.Barr) It is known that the sum of the distances from I to the vertices cannot exceed the combined distances from the orthocenter to the vertices. (Amer. Math. Monthly, E1397(1960)). Show that the reverse inequality holds for their products, namely, AH · BH · CH ≤ AI · BI · CI.

PME213.69S.S70S,84S.(G.Wulczyn) Prove that a triangle is isosceles if and only if it has a pair of equal ex-symmedian. This is false. See also Crux Math. 9 (1983) p.181. See akso MG637 for the corresponding involving symmedians.

PME215.69S.S70S.(Bankoﬀ) In an acute triangle ABC whose circumcenter is O,let YIU : Problems in Elementary Geometry 236

D, E, F denote the midpoints of sides BC, CA, AB and let P , Q, R denote the midpoints of the minor arcs BC, CA, AB of the circumcircle. Show that DP + EQ + FR sin2 A +sin2 B +sin2 C = 2 2 2 . OB + OD + OC + OE + OA + OF 2 A 2 B 2 C cos 2 +cos 2 +cos 2

PME217.69S.S70S.(C.S.Venkataraman) A transverse common tangent of two circles meets the two direct common tangents in B and C. Prove that the feet of the perpendiculars from B and C on the line of centers ar a pair of common inverse points of both the circles.

PME220.69S.S70F,71S.(Pedoe) (a) Show that there is no solution of the Apollonius problem of drawing circle to touch three given circles which has only seven solutions. (b) What specializations of the three circles will produce 0,1,2,3,4,5,6 distinct solutions ?

PME222.69F.S70F.(Garfunkel) In an acute triangle ABC, angle bisector BT1 intersects altitude AH1 in D. Angle bisector CT2 intersects altitude BH2 in E, and angle bisector AT3 intersects altitude CH3 in F .Provethat DH EH FH 1 + 2 + 3 ≤ 1. AH1 BH2 CH3

PME229.69F.S70F.(C.L.Main) Let A1 and A2 be tangent unit circles with a common external tangent T . Deﬁne a sequence of circles recursively as follows: (1) C1 is tangent to T ,A1 and A2; (2) Ci is tangent to Ci−1, A1 and A2,fori =2, 3,... Find the area of the region ∪iCi.

PME231.69F.S70F.(D.L.Silverman) (a) What is the smallest circular ring through which a regular tetrahedron of unit edge can be made to pass ? (b) What is the radius of the smallest right circular cylinder through which the unit edge tetrahedron can pass ?

PME237.70S.S71S.(L.Barr) The diameter of a semicircle is divided into two segments, a and b, by its points of contact with an inscribed circle. Show that the diameter of the inscribed circle is equal to the harmonic mean of a and b.

PME238.70S.(D.L.Silverman) A necessary and suﬃcient condition that a triangle exist is that its sides satisfy the inequalities a

Express these in a single inequality.

PME239.70S.S84F.(D.L.Silverman) A pair of tori having hole-radius = tube radius = 1 are linked. (a) What is the smallest cube into which the tori can be packed ? (b) What convex surface enclosing the linked tori has the smallest volume ? (c) What convex surface enclosing the linked tori has the smallest area ? (d) What is the locus of points in space equidistant from the two links ?

PME239’.70F.S71F.(Anonymous) Acircle(O) inscribed in a square ABCD,(AB =2a), touches AD at G, DC at F ,andBC at E.IfQ is a point on DC and P a point on BC such that GQ is parallel to AP , show that PQ is tangent to the circle (O).

2 2 2 2 2 2 9 2 2 2 2 2 2 PME242.70F.S71F.(Bankoﬀ) mamb + mb mc + mcma = 16 (a b + b c + c a ).

PME243.70F.S71F.(A.E.Neuman) Provide a geometrical proof for the well known relation π 1 1 1 =arctan +arctan +arctan . 4 2 5 8

PME247.70F.S71F.(A.E.Neuman) Construct diagrams illustrating four (or more) diﬀer- ent theorems characterised by the relation

AZ · BX · CY = |AY · BZ · CX|.

The diagrams given by the proposers invoke the following theorems: Menelaus, Ceva, De- sargues, and Morley.

PME254.71S.S72S.(A.E.Neuman) In the adjoining diagram, CD is a half-chord perpendicular to the diameter AB of a circle (O). The circles on diameters AC and CB are centered on O1 and O2 respectively. The rest of the ﬁgure consists of consecutively tangent circles inscribed in the horn - angle and in the segment as shown. If the two shaded circles are equal, what is the ratio of AC to AB ? YIU : Problems in Elementary Geometry 238

PME256.71S.S72S.(R.S.Luthar) ABCDE is a pentagon inscribed in a circle (O)with sides AB, CD and EA equal to the radius of (O). The midpoints of BC and DE are denoted by L and M respectively. Prove that ALM us ab equilateral triangle.

PME257.71S.S72S.(M.Louder and R.Field) If x, y, z are the sides of a primitive Pythagorean triangle with z>x>y,canx and x − y be the legs of another Pythagorean triangle ?

PME259.71F.S72F.(J.Bender) Prove that the product of the eccentricities of two conjugate hyperbolas is equal to or greater than 2.

PME260.71F.S72F.(Err¨os) Given n points in the plane, what is the maximum number of triangles you can form so that no two triangles have an overlapin area ?

PME269.71F.S72F.(Bankoﬀ) If A + B + C = 180◦,provethat A B C cos +cos +cos ≥ sin A +sinB +sinC. 2 2 2

PME270.72S.S73S.(Carlitz) A B C A B C cot +cot +cot ≥ 3(tan +tan +tan ) ≥ 2(sin A +sinB +sinC). 2 2 2 2 2 2

PME273.72S.S73S.(Trigg) Twelve toothpicks can be arranged to form four congruent equilateral triangles. Rearrange the toothpicks to form ten triangles of the same size.

PME277.72S.S73S.(A.E.Neuman) According to Morley’s theorem, the intersections of the adjancent internal angle trisectors of a triangle are the vertices of an equilateral triangle. If the conﬁguration is modiﬁed so that the trisectors of one of the angles are omitted, as shown in the diagram, show that the connector DE of the two intersections bisects the angle BFC.

PME288.72F.S73F.(Bankoﬀ and A.E.Neuman) If A + B + C = π, show that YIU : Problems in Elementary Geometry 239

(1) sin 2A +sin2B +sin2C ≤ sin A +sinB +sinC; (2) sin 2A +sin2B +sin2C ≤ sin A +sinB +sinC +sin3A +sin3B +sin3C; equality holding if and only if A = B = C.

PME291.72F.S73F.(Trigg) How may a square card be folded into a tetrahedron ? What is the volume of the tetrahedron in terms of the side of the square ?

PME292.73S.S74S,75S.(Garfunkel) If perpendiculars are constructed at the points of tangency of the incircle of a triangle and extended outward to equal lengths, then the join of their endpoints form a triangle perspective with the given triangle.

PME294.73S.S74S.(Trigg) Show that ABCD is a square.

PME295.73S.S74S.(Klamkin) Determine the equation of a regular dodecagon (the extended sides are not to be included).

PME300.73S.S74S.(Bankoﬀ) It can be shown without diﬃculty that if the opposite anlges of a skew quadrilateral are equal in pairs, the opposite sides are also equal in pairs. If two opposite sides of a skew quadrilateral are equal and the other two unequal, is it possible to have one pair of opposite angles equal ?

PME302.73S.S74S.(D.L.Silverman and A.E.Neuman) A tapestry is hung on a wall so that its upper edge is a units and its lower edge b units above the observer’s eye level. Show√ that in order to obtain the most favorable view the observer should stand at the distance ab from the wall.

PME305.73F.S74F.(Garfunkel) In an acute triangle ABC, AF is an altitude and P is a point on AF such that AP =2r,wherer is the inradius of the triangle. If D and E are the projections of P upon AB and AC respectively, show that the perimeter of triangle ADE is equal to that of the triangle of least perimeter that can be inscribed in triangle ABC. YIU : Problems in Elementary Geometry 240

PME311.73F.S74F.(Trigg) On opposite sides of a diameter of a circle with radius a + b two semicircles with radii a and b form a continuous curve that divides the circle into two tadpole-shaped parts. (a) Find the angle that the join of the centroids of the two component parts makes with the given diameter of the circle. (b) For what ratios a : b does the continuous curve pass through one of the centroids ? (c) When a = b, ﬁnd the moment of inertia of one of the component areas about an axis through its centroid and perpendicular to its plane.

PME313.73F,74F(corrected),S75S.(Klamkin) Give an elementary proof that

(1 + 8 cos2 A)(1 + 8 cos2 B)(1 + 8 cos2 C) ≥ 64 sin2 A sin2 B sin2 C, where A, B, C are the angles of an acute triangle ABC.

PME314.74S.S75S.(J.A.H.Hunter) Show that

sin2 45◦ − sin2 15◦ sin 80◦ = . sin2 30◦ − sin2 10◦ sin 30◦

PME316.74S.S75S.(Z.Katz) Which is greater: √ 1 5 2arctan( 2 − 1) or 3 arctan +arctan ? 4 99

PME317.74S.S75S.(Bankoﬀ) A rectangle ADEB is constructed externally on the hypotenuse AB of a right triangle ABC. The line CD and CE intersect the line AB in the points F and G respectively.√ (a) If DE = AD 2, show that AG2 + FB2 = AB2. (b) If AD = DE, show that FG2 = AF · GB.

PME319.74S.S75S.(M.S.Longuet-Higgins) Let A,B,C be the images of an arbitrary point in the sides BC. CA and AB of triangle ABC. Prove that the 4 circles ABC, BCA, CAB and ABC are all concurrent.

PME320.74S.S75S.(Coxeter) Prove that the projectivity ABC∧¯BCD for 4 collinear points is of the period 4 if and only if H(AC, BD).

PME322.74S.S75S.(Garfunkel) It is known that the ratio of the perimeter of a triangle to the sum of its altitudes is greater than or equal to √2 . (See E1427, (1961) pp.296–297). Prove 3 YIU : Problems in Elementary Geometry 241

the stronger inequality for the internal angle bisectors ta,tb,tc: √ 2(ta + tb + tc) ≤ 3(a + b + c), equality holding if and only if the triangle is equilateral.

PME323.74S.S75S.(D.L.Silverman) Call plane curves such as the circle of radius 2, the square of side 4, or the 6 × 3 rectangle isometric if theri perimeter is numerically equal to the area they enclose. What is the maximum area that can be enclosed by an isometric curve ?

PME331.74F.S75F.(Garfunkel) In a right triangle ABC, A =60◦, B =30◦,withD, E, F the points of trisection nearest A, B, C on the sides AB, BC, CA respectively. Extend CD,AE and BF to intersect the circumcircle (O)atpointsP, Q, R. Show that triangle PQR is equilateral.

PME335.74F.S75F.(V.G.Feser) (a) Show that every simple polygon of n sides, n ≥ 3, has at least n − 3 interior diagonals. (b) SHow that for every n ≥ 3, there existsa simple polygon having exactly n − 3interior diagonals. See also PME65.54S.S69F.

PME336.74F.S75F.(Z.Katz) On the diameter AB of a semicircle (O) perpendiculars are erected at arbitrary points C and D cutting the semi-circumference at points E and F respectively. A circle (P ) touches the arc of the semicricle and each of the two half-chords. Show that PQ, the distance from P to the diameter AB, is equal to the geometric mean of AC and DB.

PME337.74F.S75F.(Bankoﬀ) If R, r and ρ denote the circumradius, the inradius and the orthic triangle radius respectively of an acute triangle, show that r2 ≥ ρR.

PME338.75S.S76S.(H.C.Li) Let (O)a be a circle centered at O with radius a.LetP ,any point on the circumference of (O), be the center of circle (P ). What is the radius of (P )such that it divides the area of (O) into two regions whose areas are in the ratio s : t ? YIU : Problems in Elementary Geometry 242

PME341.75S.S76S,76F.(Garkfunkel) Prove that the following construction triects an angle of 60◦. Triangle ABC is a 30◦ − 60◦ − 90◦ right triangle inscribed in a circle. Median CM is drawn to side AB and extended to M on the circle. Using a marked straightedge, point N on AB is located such that CN extended to N on the ecircle makes NN = MM.ThenCN trisects the 60◦ angle ACM.

PME344.75S.S76S.(J.A.H.hunter) Three circles whose radii are a, b and c are tangent externally in pairs and are enclosed by a triangle each side of which is an extended tangent of two of the excircles. Find the sides of the triangles.

PME346.75S.S76S.(R.S.Luthar) The internal angle bisectors of a convex quadrilateral ABCD enclose another quadrilateral EFGH.LetFE and GH meet in M and let GF and HE meet in N. If the internal bisectors of angles EMH and ENF meet in L, show that angle NLM is a right angle.

PME351.75F.S76F.(Garfunkel) Angles A and B are acute angles in ABC.IfA =30◦ and ha, the altitude issuing from A,isequaltomb, the median issuing from B, ﬁnd angles B and C.

PME352.75F.S76F.(Trigg) The edges of a semi-regular polyhedron are equal. The faces consist of eight equilateral triangles and six regular octagons. In terms of the edge e, ﬁnd the diameters of the following spheres: (i) the sphere touching the octagonal faces, (b) the circumsphere, and (c) the sphere touching the triangular faces. See also PME198.67S.S68F.

PME354.75F.S76F.(A.Bernhart and D.C.Kay) In ABC with angles less than 60◦, the Fermat point, deﬁned as that point which minimizes the function f(X)=AX + BX + CX, may be determined as the point P of concurrence of lines AD, BE, CF ,whereBCD, ACE and ABF are equilateral triangles constructed externally on the sides of the triangle ABC.IfR, S and T are the points where PD, PE,andPF meet the sides of triangle ABC,provethatPD, PE and PF are twice the arithmetic means, and that PR, PS and PT are half the harmonic means of the pairs of distances (PB,PC), (PC,PA), and (PA,PB) respectively.

PME361.75F.S76F.(C.A.Argila) Consider any triangle ABC such that the midpoint P of side BC is joined to the midpoint Q of side AC by the line segment PQ. Suppose R and S are the projections of P and Q respectively on AB, extended if necessary. what relationship must hold between the sides of the triangle if the ﬁgure PQRS is a square.

PME362.76S.S77S.(Z.Katz) AdiameterAB of a circle (O) passes through C, the midpoint YIU : Problems in Elementary Geometry 243 of a chord DE. M is the midpoint of arc AB and the chord MP passes through C. The radius OP cuts the chord DE at Q. The tangent circle (O)1), (O2), (W1), (W2) are drawn as shown. Show that DQ = W1W2.

PME367.76S.S77S.(R.R.Rowe) A box of unit volume consists of a square prism topped by a pyramid. Find the side of the square base and heights of prism and pyramid to minimize the surface area.

PME368.76S.S77S.(Garfunkel) Given a triangle ABC with its inscribed circle (I), lines AI, BI, CI cut the circle in points D, E, F respectively. Prove that AD + BE + CF ≥ √1 3 perimeter of DEF.

PME374.76F.S77F.(Garfunkel) In a triangle ABC inscribed in a circle (O), angle bisectors AT1, BT2, CT3 are drawn and extended to the circle. Perpendicualrs T1H1, T2H2, T3H3 are drawn to sides AC, BA, CB respectively. Prove that T1H1 + T2H2 + T3H3 does not exveed 3R,whereR is the circumradius.

PME380.76F.S77F.(V.F.Ivanoﬀ) Form a square from a quadrangle by bisecting segments and the angles.

PME383.76F.S77F.(N.Schaumberger) Find a pentagon such that the sum of the squares of its sides is equal to four times its area.

PME385.76F.S77F.(J.T.Hurt) Solve sin α =tan(α − β)+cosα tan β.

PME386.77S.S78S.(Trigg) Show that the volume of Kepler’s Stella Octagula ( a compound of two interpenetrating tetrahedra) is three times that of the octahedron that was stellated. YIU : Problems in Elementary Geometry 244

PME387.77S.S78S.(Garfunkel) On the sides AB and AC of an equilateral triangle ABC mark the points D and E respectively such that AD = AE. Erect equilateral triangles on CD, AE and AB, as in the ﬁgure, with P, Q, R as the respective third vertices. Show that triangle PQR is equilateral. Also show that the midpoints of PE,AQ and RD are vertices of an equilateral triangle.

PME390.77S.S78S.(R.Koether and D.C.Kay) Let the diagonals of a regular n−gon of unit side be drawn. Prove that the n − 2 consecutive triangles thus formed which have their bases along one diagonal, their legs along two others or a side, and one vertex in common with a vertex of the polygon each have the property that the product of two sides equals the third.

PME394.77S.S78S.(E.Just and B.Kabak) 3(sin2 A+sin2 B+sin2 C)−2(cos3 A+cos3 B+ cos3 C) ≤ 6.

PME398.77S.S78S.(R.S.Field) Find solutions in integers A = B = C = R and A = B = C = R for the quadrilateral inscribed in a semicircle of radius R, as shown in the diagram below. Find also solutons in integer A = B = C = R or prove that none exist.

PME399.77F.S78F.(Garfunkel) Show that x − 3 x π arcsin + 2 arccos = , 3 ≤ x ≤ 6. 3 6 2

PME405.77F,78F(corrected).S79F.(N.Schaumberger) Locate a point P in the interior of a triangle such that the product of the three distances from P to the sides of the triangle is amaximum.

PME406.77F.S78F.(Erd¨os) Let there be given 5 distinct points in the plane. Suppose they determine only two distances. Is it true that they are the vertices of a regular pentagon ?

PME408.77F.S78F.(C.W.Dodge) Squares are erected on the sides of a triangle, either all externally or all internally. A circle is centered at the center of each square wotj each radius a YIU : Problems in Elementary Geometry 245

ﬁxed multiple k>0 of the side of that square. Find k so that the radical center of the three circles falls on the Euler line of the triangle and ﬁnd where on the Euler line it falls.

PME409.77F.S78F.(Z.Katz) ApointE is chosen on side CD of a trapezoid ABCD, AD//BC), and is joined to A and B. A line through D parallel to BE intersects AB in F . Show that FC is parallel to AE.

PME410.77F.S78F.(Klamkin) If x, y, z are the distances of an interior point of a triangle ABC to the sides BC, CA, AB, show that 1 1 1 2 + + ≥ x y z r where r is the inradius of the triangle.

PME412.78S.S79S.(S.W.Golomb) Are there examples of angles which are trisectible but not constructible ? That is, can you ﬁnd an angle α which is not constructible with straight α edge and compass, but such that when α is given, 3 can be constructed from it with straight edge and compass ?

PME416.78S.S79S.(S.Kim) Each of the three ﬁgures shown above is composed of two isosceles right triangles, ABC and DBE,where ABC and DBE are right angles, and B is between points A and D. CB In (a), points C and E coincide, so that EB =1. CB In (b), EB =2. CB In (c), EB =3. Consider each pair of triangles asa single shape and suppose that the areas of the three shapes are equal.

Problem: for each pair of figures, find the minimum number of pieces into which the first figure must be cut so that the pieces may be reassembled to form the second figure. Pieces may not overlap, and all pieces mut be used in each assembly. YIU : Problems in Elementary Geometry 246

PME417.78S.S79S.(C.W.Dodge) (a) Prove that the line joining the midpoints of the diagonals of a quadrilateral circumscribed about a circle passes through the center of the circle. (b)LettheincircleoftriangletouchsideBC at X. Prove that the line joining the midpoints of AX and BC passes through the incenter I of the triangle.

PME418.78S.S79S.(R.C.Gebhardt) Find all angles θ such that tan 11θ = tan 111θ = tan 1111θ = tan 11111θ = ···.

PME420.78S.S79S.(H.Taylor) Given four lines through a point in 3-space, no three of the lines in a plane, ﬁnd four points, one on each line, forming the vertices of a parallelogram. See also Putnam Competition, 1977, B2.

PME421.78S.S79S.(Klamkin) If F (x, y, z) is a symmetric increasing function of x, y, z, prove that for any triangle, in which wa,wb,wc are the internal angle bisectors and ma,mb,mc themedians,wehave F (wa,wb,wc) ≤ F (ma,mb,mc) with equality if and only if the triangle is equilateral.

PME422.78S.(Garfunkel) If perpendiculars are erected outwardly at A, B of a right triangle ABC, C =90◦), and at M, the midpoint of AB, and extended to points P , Q,andR such 1 that AP = BQ = MR = 2 AB, show that triangle PQR is perspective with triangle ABC.

PME425.78F.S79F.(Trigg) Without using its altitude, compute the volume of a regular tetrahedron by the prismoidal formula. √ PME427.78F.S79F.(J.E.Fritts)√ If a, b, c, d are integers and u = a2 + b2, v = (a − c)2 +(b − d)2, and w = c2 + d2,then (u + v + w)(u + v − w)(u − v + w)(−u + v + w) is an even integer. Solution. This is four times the area of the triangle with vertices (a, b), (c, d), and the origin, and is therefore |ad − bc|.

PME428.78F.S79F.(S.W.Golomb) One circle of radius a may be “exactly surrounded” by 6 circles of radius a. It may also be exactly surrounded by n circle of radius t, for any n ≥ 3, where π t = a(csc − 1)−1. n YIU : Problems in Elementary Geometry 247

Suppose instead we surround it with n + 1 circles, one of radius a and n of radius b, (again ≥ b n 3). Find an expression for a as a fucntion of n.

PME430.78F.S79F.(J.M.Howell) Given any rectangle, form a new rectangle by adding a square to the long side. What is the limit of the long side to the short side ?

PME431.78F.S79F.(Garfunkel) In a right triangle ABC, with sides a, b and hypotenuse c, show that 4(ac + b2) ≤ 5c2.

PME435.78F.S79F.(D.R.Simonds) Two non-congruent triangles are “almost congruent” if two sides and three angles of one triangle are congruent to two sides and three angles of the other triangle. Clearly two such√ triangles are similar. Show that the ratio of similarity k is such 1 1 that φ

PME436.78F.S79F.(C.Spangler and R.A.Gibbs) P1 and P2 are distinct points on lines L1 and L2 respectively. Let L1 and L2 rotate about P1 and P2 respectively with equal angular velocities. Describe the locus of their intersection.

PME437.78F.S79F.(Z.Katz) In times gone by, it was fairly well known that N, the Nagel point of a triangle, is the intersection of the lines from the vertices to the points of contact of the opposite escribed circles. In the triangle whose sides are AB =5,BC =3andCA =4, show that the areas of triangles ABN, CAN and BCN are 1,2,3 respectively.

PME438.79S.S80S,80F,81S.(E.Straus) Prove that the sum of the lengths of alternate sides of a hexagon with concurrent major diagonals inscribed in the unit circle is less than 4.

PME442.79S.S80S.(Garfunkel) Show that the sum of the perpendiculars from the circumcenter of a triangle to its sides is not less than the sum of the perpendiculars drawn from the incenter to the sides to the triangle.

PME447.79S.S80S.(Z.Katz) A variable circle touches the circumferences of two internally tangent circles, as shown in the ﬁgure. YIU : Problems in Elementary Geometry 248

(a) Show that the center of the variable circle lies on an ellipse whose foci are the centers of the ﬁxed circles. (b) Show that the radius of the variable circle bears a constant ratio to the distance from its center to the common tangent of the ﬁxed circles. (c) Show that this constant ratio is equal to the eccentricity of the ellipse.

PME448.79S.S80S.(R.R.Rowe) Analogous to the median, call a line from a vertex of a triangle to a third point of the opposite side a “tredian”. Then if both tredians are drawn from each vertex, the 6 lines will intersect at 12 interior points and divide the area into 19 subareas, each a rational part of the area of the triangle. Find two triangles for which each subarea is an integer, one being a Pythagorean right triangle and the other with consecutive integers for its three sides.

PME450.79F.S80F.(C.W.Dodge) In triangle ABC, A ≤ B ≤ C.Provethat √ s>, =or< 3(R + r) if and only if B>, =or< 60◦.

PME453.79F.S80F.(Garfunkel) Given two intersecting lines and a circle tangent to each of them, construct a square having two of its vertices on the circumference of the circle and the other two on the intersecting lines.

PME454.79F.S80F.(M.Haste) The point within a triangle whose combined distances to the vertices is a minimum is known as the Feremat - Toricelli point, designated by T .Ina triangle ABC,ifAT , BT, CT form a geometric progression with a common ratio 2, ﬁnd the angles of the triangle.

PME459.79F.S80F.(B.Prielipp) If (x, y, z) is a Pythagorean triple in which x and z are prime numbers and x ≥ 11, show that 60 divides y.

PME460.79F.S80F.(B.Seville) The dihedral angle of a cube is 90◦. The other four Platonic solids have dihedral angles which are approximately 70◦3143.60, 109◦2816.3956, 116◦3354.18, and 138◦1122.866. How closely can these angles be constructed with straightedge and compass ? Can good approximations be accomplished by paper folding ? If so, how ?

PME461.79F.S80F.(D.C.Kay) (a) A right triangle with unit hypotenuse and legs r and s is used to form a sequence of similar right triangles T1,T2,T3,... where the sides of T1 are r times those of the given triangle, and for n ≥ 1 the sides of Tn+1 are s times those of Tn.Prove that the sequence Tn will tile the given triangle. (b) What happens if the multipliers r and s are reversed ? YIU : Problems in Elementary Geometry 249

(c) The art of the Hopi American Indians is known for its zigzag patterns. The blanket illustrated below is made from a rectangle of (inside) dimensions a × b,andthezigzagisformed by dropping perpendiculars to alternating sides of the triangles in the design. Show that the a3b+ab3 area of the design (shaded portion) is given by the formula 2a2+4b2 .

PME465.80S.S81S.(Trigg) What is the shortest stripof equilateral triangles of side k that, while remaining intact, can be folded along the sides of the triangles so as to completely cover the surface of an octahedron with edge k ?

PME469.80S.S81S.(R.I.Hess) Start with a unit circle and circumscribe an equilateral triangle about it. Then circumscribe a circle about the triangle and a square abotu the circle. Continue indeﬁnitely circumscribing circle, regular pentagon, circle, regular hexagon, etc. (a) Prove that there is a circle which contains the entire structure. (b) Find the radius of the smallest such circle.

PME471.80S.S81S.(C.W.Dodge) Let two circles meet at O and P ,and let the diameters OS and OT of the two circles cut the other circle at A and B. Prove that chord OP passes through the cneter of circle OAB.

PME473.80S.S81S.(Garfunkel) In an acute triangle with angle A =60◦, P is a point within the triangle. D and E are the feet of the cevians through P ,fromC and B respectively. (a) If BD = DE = EC,provethatAP = BP = CP. (b) Conversely, if AP = BP = CP,provethatBD = DE = EC. (c) If PBC = PCB =30◦, show that BD = DE = EC.

PME475.80F.S81F.(Z.Katz) In the accompanying diagram, DC is the radius perpendicular to the diameter AB of the semicircle ADB; FG is a half-chord parallel to DC; AF cuts DE DC in E. Show that the sides of triangle FCG are integers if and only if EC or its reciprocal is an integer.

PME476.80F,(correction 81S).S82S.(Garfunkel) If A, B, C, D are the internal angles of YIU : Problems in Elementary Geometry 250 a convex quadrilateral, then √ A B C A B C 3(cos +cos +cos ) ≤ cot +cot +cot , 2 2 2 2 2 2 with equality when A = B = C = D =90◦.

PME484.80F.S81F.(R.R.Robinson) In a triangle with base AB and vertex C, secants from A and B to points D and E on BC and CA divide the area into four subareas S, T , U and V . In some order of S, T , U, V ,thepointsD and E can be located so that the subareas are in increasing arithmetical progression, or so that they are in decreasing arithmetical progression. Find that order and evaluate the subareas.

PME485.80F.S81F.(R.S.Luthar) A line l cuts two parallel rays emanating from L andM in A and B respectively. A point C is taken anywhere on l.LinesthroughA and B respectivley parallel to MC and LC intersect in P . Find the locus of P .

PME492.81S.S82S.(Garfunkel) Given an acute triangle ABC with altitudes ha, hb, hc, and medians ma, mb, mc.ThepointsP , Q, R are the intersections of ma and bb; mb and hc; and mc and ha respectively. Show that AP BQ CR + + ≥ 6. PD QE RF Here, D, E, F are the midpoints of the sides.

PME494.81S.S82S.(Z.Katz) In the diagram CD is a half-chord perpendicular to the diameter AB of the semicircle (O),andtheinscribedcircle(P )touchesAB in J and the arc DB in K.Show by elementary plane geometry, without using inversion, that

PME495.81S.S82S.(R.Hess) A regular pentagon is drawn on an ordinary graph paper. Prove that no more than two of its vertices lie on grid points.

PME496.81S.S82S.(D.Conrad) P is any point within ABC.Ifx is the distance from P to BC, show that ax PA2 = PH2 + b2 + c2 − 4R2 − (b2 + c2 − a2). 2s

PME501.81F.S82F.(R.C.Gebhardt) A rectangle is inscribed inside a circle. The area of the circle is twice the area of the rectangle. What are the proportions of the the rectangle ? YIU : Problems in Elementary Geometry 251

√ Answer. 4 − 16 − π2 : π ≈ 0.485.

PME507.81F.S82F.(H.R.Bailey) A unit square is to be covered by three circles of equal radius. Find the minimum necessary radius. See also PME690.88F.

PME503.81F.S82F.(Garfunkel) Given a triangle ABC whose incircle touches the sides BC, CA, AB at L, M, N.LetP , Q, R be the midpoints of the arcs NL, LM and MN respectively. Form triangle DEF by drawing tangents to the circle at P , Q, R. Prove that the perimeter of DEF ≤ perimeter of ABC.

PME514.82S.S83S.(R.E.Spaulding) Let A1A2 ···An be a regular n−gon each side of length 1. Let Bi be a point on AiAi+1 such that AiBi = x.LetCi be the point where AiBi+1 intersects Ai+1Bi+2. Find the area of the regular polygon C1C2 ···Cn in terms of n and x.

PME515.82S.S83S.(Garfunkel) Given a sequence of concentric circles with a triangle ABC circumscribing the outermost circle. Tangent lines are drawn from each vertex of ABC to the next inner circle, forming the sides of triangle ABC. Tangents are now drawn from vertices A, B, C to the next inner circle and they are the sides of triangle ABC and so on. Prove (n) (n) (n) π that the angles of triangle A B C approach 3 .

PME523.82F.S83F.(Rabinowitz) Let ABCD be a parallelogram. Eerect directly similar right triangles ADE and FBA outwardly on sides AB and DA (so that ADE and FBA are right angles. Prove that CE and CE are perpendicular.

PME528.82F.S83F.(A.Wayne) Call a trio like (19, 24, 35), (15, 29, 34) and (14, 31, 33) a size triplet because the three triangles have the same perimeter and the same area. Since the common area is least, this is the smallest size triplet. What is the next larger size triplet ? Remark: See also AMME2872.S827. Example of 10 triangles with equal area and perimeter:

(124700, 830280, 579020), (1246032, 752250, 653718), (1245675, 765765, 640560), (1182675, 1101360, 367965), (1186770, 1093950, 371280), (1206660, 1047540, 397800), (1219920, 1001130, 430950), (1233180, 928200, 490620), (1236495, 901680, 513825), (1246440, 729300, 676260).

PME530.82F.S83F.(Bankoﬀ) In the accompanying diagram, AB =2r is the diameter of circle (O)andAC =2r1 the diameter of circle (O1), D is a point on diameter AC,andthehalf -chordDQ perpendicular to AC cuts the circle (O1)atP .Thecircle(W1)ofradiusρ1 and YIU : Problems in Elementary Geometry 252

(W2)ofradiusρ2 are tangent to circle (O)and(OP1) and touch PQ on opposite sides. Show ρ1 r1 that ρ2 = r .

PME540.83S.S84S.(Klamkin) If the exradii satisfy (r1 − r2)(r1 − r3)=2r2r3, determine which of the angles A, B, C is the largest.

PME541.83S.S84S.(Rabinowitz) A line meets the boundary of an annulus A1 in four points P , Q, R, S with R and S between P and Q. A second annulus A2 is constructed by drawing circles on PQ and RS as diameters. Find the relationshipbetween the areas of A1 andA2.

PME542.83S.S84S.(H.R.Bailey) A circle of unit radius is to be covered by three circles of equal radii. Find the minimum radius required.

PME544.83S.S84S.(Garfunkel) Show that a quadrilateral ABCD with sides AD = BC = s and A + B = 120◦ has maximum area if it is an isosceles trapezoid.

PME550.83F.S84F.(I.R.Hess) How many diﬀerent Pythagorean triples have a side or hypotenuse equal to 1040 ?

PME553.83F.S84F.(Garfunkel) Given a triangle ABC rect equilateral triangles BAP, ACQ outwardly on sides AB and CA.LetR be the midpoint of side BC and let G be the centroid of triangle ACQ. Prove that triangle PRG is a 30◦ − 60◦ − 90◦ triangle.

PME555.83F.S84F.(R.D.Stratton) 18 toothpicks can be arranged to form six congruent equilateral triangles. Rearrange the toothpicks to form sixteen congruent equilateral triangles each of the same size as the original six.

PME558.83F.S85F.(R.I.Hess) Let ABCD be a quadrilateral. Let each of the sides AB, BC, CD, DA be the diagonal of a square. Let E, F . G. H be those vertices of the squares that lie outside the quadrilateral. Prove that EG and FH are perpendicular. YIU : Problems in Elementary Geometry 253

PME560.83F.S84F.(Bankoﬀ) In a given circle the radii OA and OB are perpendicular. Let the circle on OB as diameter have center O and let OA cut this new circle in point D. Then AD is the length of the side of a regular decagon inscribed in the given circle. Also, let the tangent AQ to the new circle cut the given circle again at P .ThenAP is the length of the side of a regular pentagon inscribed in the given circle.

PME562.84S.S85S.(W.Blumberg) Prove that tan 1◦ tan 61◦ =tan3◦ tan 31◦.

PME565.84S.S85S.(W.Blumberg) Let ABCD beasquareandchooseapointE on segment AB and point F on segment BC such that angles AED and DEF are equal. Prove that EF = AE + FC.

PME567.84S.S85S.(R.S.Luthar) Findtheexactvalueofsin20◦ sin 40◦ sin 80◦.

PME569.84S.S85S.(R.C.Gebhardt) (a) Find the largest regular tetrahedron that can be folded from a square piece of paper (with cutting). (b) Prove whether it is possible to fold a regular tetrahedron from a square piece of paper without overlapping or cutting.

PME572.84S.S85S.(Garfunkel) Let ABCD be a parallelogram and construct directly similar triangles on sides AD, BC and diagonals AC and BD. See the ﬁgure, in which triangles ADE, ACH, BDF and BCG are the directly similar triangles. What restrictions on the ap- pended triangles are necessary for EFGH to be rhombus ?

PME580.84F.S85F.(B.Prielipp) Let a, b, c be the lengths of the sides of a triangle and s its semiperimeter. Prove that a a b b c c ≥ (s − a)a(s − b)b(s − c)c. 2 2 2

PME581.84F.S85F.(Rabinowitz) If a triangle similar to a 3-4-5 triangle has its vertices at lattice points in the plane, must its legs be parallel to the coordinate axes ? YIU : Problems in Elementary Geometry 254

PME582.84F.S85F.(W.Blumberg) Suppose bc2 cos B = ca2 cos C = ab2 cos A.Provethat the triangle is equilateral.

PME584.84F.S85F.(Garfunkel) Let ABC be any triangle with base BC.LetD be any point on side AB and E any point on side AC.LetPDE be an isosceles triangle with base DE,orientedthesameasABC, and with apex angle P equal to angle A. Find the locus of all such points P .

PME596.85S.S86S.(Rabinowitz) Two circles are externally tangent and tangent to a line L at points A and B. A third circle is inscribed in the curvilinear triangle triangle bounded by these two circles and L and it touches L at point C. A fourth circle is inscribed in the curvilinear triangle bounded by line L and the circles at A and C and it touches the line at D.Findthe relationshipbetween the lengths AD, DC and CB.

PME597.85S.S86S.(Rabinowitz) Find the smallest n such that there exists a polyhedron of nonzero volume and with n edges of lengths 1,2, . . . , n.

PME602.85F.S86F.(Garfunkel) Given isosceles triangle ABC and a point O in the plane of the triangle, erect directly similar isosceles triangles POA, QOB and ROC (but not necessarily similar to ABC). Prove that the apexes P , Q, R of these triangles determine a triangle similar to ABC.

PME604.85F.S86F.(D.Iny) A unit square is covered by n congruent equilateral triangles of side s without the triangles overlapping each other. Find the minimum value of s for n =1, 2, 3.

PME607.85F.S86F.(Garfunkel) Triangles ABC and ABC are right triangles with right s s s s angles at C and C .Provethatifr = r ,then R < R .

PME616.86S.S87S.(D.P.Mavlo) Prove that in any triangle tan A +tanB +tanC 8 A B C 2 2 2 2 ≤ + tan tan tan , A B C 27 2 2 2 cot 2 +cot 2 +cot 2 with equality if and only if the triangle is equilateral.

PME620.86S.(Garfunkel) A triangle ABC isinscribed in an equilateral triangle PQR.The angle bisectors of triangle ABC are drawn and extended to meet the sides of triangle PQR in points A1,B1,C1. Now draw the angle bisectors of A1B1C1 to meet the sides of triangle PQR at A2, B2, C2. Repeat the procedure. Prove or disprove that triangle AnBnCn tends to YIU : Problems in Elementary Geometry 255 equilateral as n tends to inﬁnity. (This result has been proved when a circle is used instead of triangle).

PME621.86S.S87S.(R.S.Luthar) (i) Characterize all triangles whose angles and whose sides are both in arithmetic progression. (ii) Characterize all triangles whose angles are in arithmetic progression and whose sides are in geometric progression.

PME622.86S.(W.Blumberg) Let P be the cneter of an equilateral trianlge ABC and let C be any circle centered at P and lying entirely within the triangle. Let BR and CS be tangents to the circle such that point R is closer to C than to A and S is closer to A than to B.Prove that line RS bisects side BC.

PME629.86F.S87F.(Garfunkel) Prove that

cos A cos B cos C ≤ (1 − cos A)(1 − cos B)(1 − cos C).

PME630.86F.S87F.(R.Euler) Evaluate

j mπ sin . m=1 2j +1

PME637.86F.S87F.(R.S.Luthar) Let ABC be a triange with ABC = ACB =40◦.Let BD be the bisector of ABC and produce it to E so that DE = AD. Find the measure of BEC.

PME638.86F.S87F.(R.S.Luthar) The circle with center O is an excircle of triangle ABC. Then BK is drawn so that KBA = AOC,andOA is produced to meet BK in D.Prove that OCBD is a cyclic quadrilateral. YIU : Problems in Elementary Geometry 256

PME644.87S.S88S.(R.I.Hess) In the ﬁgure below, prove that regions A and B have equal areas.

PME645.87S.S88S.(D.P.Mavlo) Let M be an arbitrary point on segment CD of trapezoid ABCD having sides AD and BC parallel. Let S, S1,andS2 be the areas of triangles ABM, BCM,andADM.Provethat S ≥ 2min(S1,S2).

PME648.87S.S88S.(Garfunkel) In any triangle ABC,provethat √ A 3 A cos ≤ cos2 . 2 6 2

PME653.87F.S88F.(R.C.Gebhardt and C.H.Singer) A small square is constructed in- 1 side a square of area 1 by marking oﬀ segments of length n along each side as shown below. For 1 n = 4, the side s of the small square is 5 . For what other values of n is s the reciprocal of an integer ?

◦ PME655.87F.S88F.(R.S.Luthar)√ IN ABD, B = 120 . There is a popint C on the side ◦ 3 2 AD such that ABC =90 , AC = 2, and BD = AC . Find the lengths of AB and CD.

PME656.87F.S88F.(Garfunkel) Let ABC be any triangle and extend side AB to A,side BC to B, and side CA to C so that B lies between A andA etc., and BA = λAB, AC = λCA, YIU : Problems in Elementary Geometry 257 and CB = λBC. Find the value of λ that the area of triangle ABC is 4 times the area of ABC.

PME657.87F.S88F.(R.S.Luthar) Evaluate π 3π 5π 7π sin6 +sin6 +sin6 +sin6 . 8 8 8 8

PME661.87F.S88F.(J.M.Howell) (a) How close to a cubical box can you get if the sides and the diagonal of a rectangular parallelepiped are all integral ? (b) How close can you get to a cube if all the face diagonals must be integral too ?

PME673.88S.S89S.(Rabinowitz) Let AB be an edge of a regular tessereact (a four- dimensional cube) and let C be the tesseract’s vertex that is furthest from A. Find the measure of anlge ACB.

PME675.88S.S89S.(J.H.Scott) Erect a semicircle on a segment AB as diameter. From point D on the semicircle drop a perpendicular to point C on AB. Draw a circle tangent to CB at J and tangent to the semicircle and to segment CD. Prove that angles CDJ and JDB have equal measures.

PME677.88S.S89S.(Garfunkel) In any triangle ABC, √ cos A cos B cos C 3 ≤ . A B C 9 cos 2 cos 2 cos 2

PME683.88F.S89F.(Garfunkel) (a) Given three concentric circles construct an isosceles right triangle so that its vertices lie one on each circle. (b) Is the construction always possible ? YIU : Problems in Elementary Geometry 258

PME684.88F.S89F.(D.P.Mavlo) Erdös and Hans Debrunner, Elem. der Math. 11 (1956) p.20), proved the following theorem: Let D, E, F be points on the interiros of sides BC, CA, AB of triangle ABC. Then the area of DEF cannot be less than the smallest of the three other triangles formed: [DEF] ≥ min([AEF ], [CDE], [BFD]). (a) Prove this generalization of the Erdös - Debrunner theorem: for some fixed real number α,if−∞

1 [AEF ]t +[CDE]t +[BFD]t t [DEF] ≥ . 3 (b) Determine all the cases where equality holds. (c) Prove that for t = −1, the inequality of part (a) is equivalent to the inequality 1 1 1 (1 + xyz) + + ≥ 3, x(y + z) y(z + x) z(x + y) with equality if and only if x = y = z.

PME685.88F.S89F.(R.S.Luthar) In any triangle ABC with C<45◦, and given any other angle D with 0◦

PME690.88F.S89F.(D.Iny) A unit square is covered by 5 circle of equal radius. Find the minimum necessary radius. See also PME507.82F.

PME695.89S.S90S.(Garfunkel) In any triangle ABC,provethat √ √ √ √ A B C sin A + sin B + sin C ≥ 6 3sin sin sin . 2 2 2

PME697.89S.S90S.(K.Goggin) Circle (B) is internally tangent to circle (A)atK and to diameter VAW at cneter A.Circle(C) is internally tangent to circle (A)atZ, externally tangent circle (B)atL. Find the ratios of the areas of the three circles to one another. √ PME701.89S.S90S.(D.P.Mavlo)√ Let L and B be nonnegative numbers such that 3L + 9B =9 3. Prove that in any triangle ABC, tan A +tanB +tanC A B C 2 A B C 2 2 2 ≥ L tan tan tan , +B tan tan tan . A B C 2 2 2 2 2 2 cot 2 +cot 2 +cot 2 YIU : Problems in Elementary Geometry 259 with equality if and only if the triangle is equilateral.

PME702.89S.S90S.(D.P.Mavlo) In right triangle ABC with right angle at C the altitude CD and the median CE are drawn. Determine the ratio of the sides containing the right angle if AB =3DE.

PME707.89F.S90F.(Klamkin) From a point R taken on any circular arc PQ of less than a quadrant, two segments are drawn, one to an extremity P of the arc and the other RS perpendicular to the chord PQ of the arc and terminated by it. Determine the maximum of the sum PR+ RS of the lengths of the two segments. This problem without solution is given in Todhunter’s Trigonometry.

PME708.89F.S90F.(Garfunkel) Find a Mascheroni construction (a construction using only compass, no straightedge allowed) for the orthic triangle of an acute triangle.

PME709.89F.S90F.(N.Schaumberger) In any triangle ABC, 1 a2b2 + b2c2 + c2a2 ≥ 122 + s4. 8

PME711.89F(correction, 90F).S90F.(J.N.Boyd) A pentagon is constructed with ﬁve segments of lengths 1,1,1,1, and w.Findw so that the pentagon will have the greatest area.

PME724.90S.S91S.(Klamkin) Which of the following triangle inequalities, if any, are valid ? (a) max(ha,hb,hc) ≥ min(ma,mb,mc); (b) max(wa,wb,wc) ≥ min(ma,mb,mc); (c) mid (wa,wb,wc) ≥ min(ma,mb,mc).

PME727.90S.S91S.(Garfunkel) In any triangle ABC,provethat B − C 2+ cos ≥ 2 cos A. 2

PME729.90S.S91S.(D.P.Mavlo) Given a non-obtuse triangle with altitude CD = h, drawn to side AB, denote the inradii of triangles ACD BCD and ABC by r1, r2 and r3 respectively. Prove that if r1 + r2 + r3 = h, then the triangle is right angled at C.

PME731.90S.S91S.(Garfunkel) (a) Show that on the lattice points in the plane one cannot have the vertices of an equilateral triangle. YIU : Problems in Elementary Geometry 260

(b) What about a tetrahedron in space ?

PME741.90F.S91F.(J.M.Howell) (a) What numbers cannot be a leg of a Pythagorean triangle ? (b) What numbers cannot be a hypotenuse of a Pythagorean triangle ? (c) What numbers can be neither a leg nor a hypotenuse of a Pythagorean triangle ?

PME742.90F.S91F.(Garfunkel) Construct squares outwardly on the sides of a triangle ABC. Prove or disprove that the centers A, B, C of these squares form a triangle that is closer than being equilateral than is ABC. A proof would show that if the process were repeataed on triangle ABC, etc., that triangle AnBnCn would approach equilateral as n approach inﬁnity.

PME743.90F.S91F.(R.S.Luthar) Let A and B be the ends of the diameter of a semicircle of radius r and let P be any point on the semicircle. Let I be the incenter of triangle AP B. Find the locus of I as P moves along the semicircle.

PME744.90F.S91F.(Garfunkel) Let ABC be inscribed in a circle. Draw a line through A to intersect side BC at D and the circle again at E. Without resorting to calculus, prove AD that DE is minimum when AD bisects angle A.

PME747.91S.S92S.(Garfunkel) Let ABC be a triangle with inscribed circle (I)andlet the line segments AI, BI, CI cut hte incircle at A, B, C respectively. Prove that A B C sin A +sinB +sinC ≥ cos +cos +cos . 2 2 2

PME755.91S.S92S.(S.Rabinowitz) In triangle ABC, a circle of radius p is inscrbed in the wedge bounded by sides AB and BC and the incircle (I) of the triangle. A circle of radius q is inscrbed in the wedge bounded by sides AC and BC and the incircle. If p = q,provethat AB = AC.

PME757.91S.S92S.(P.A.Coatney) Find the overall height of the pyramid formed from YIU : Problems in Elementary Geometry 261 four spherical balls of radius r.

PME759.91F.S92F.(J.E.Wetzel) Call a plane arc special if it has length 1 and lies on one side of a line through its endpoints. Show that anhy special arc can be contained in an isosceles right triangle of hypotenuse 1.

PME760.91F.S92F.(J.E.Wetzel) Napoleon’s Theorem is concerned with erecting equilateral triangles outwardly on the sides of a given triangle ABC.ThenDEF is the triangle formed by the third vertices of these equilateral triangles BCD, CAE,andABF . Lemoine asked in 1868 if one can reconstruct ABC when only DEF is given. Shortly afterward, Keipert showed that the construction is to erect outward equilateral triangles EFX, FDY and DEZ on triangle DEF,andthenA, B, C are the midpoints of the segments DX, EY , FZ.Hisproof was quite tedious. Find a simple proof of Keipert’s construction.

PME768.91F.S92F.(Garfunkel) Given triangle ABC, draw rays inwardly from each vertex to form a triangle ABC such that B, C, A lie on rays AA, BB, CC respectively, and

BAB = ACA = CBC = α, as shown in the ﬁgure. Prove that (a) ABC is similar to ABC; (b) the ratio of similitude is cos α − sin α cot ω,whereω is the Brocard angle of triangle ABC.

Solution. (a) Note that BAC = AAC + ACA = AAC + α = AAC + AAB = BAC. Similarly, CAB = CAB. It follows that the triangles ABC and ABC are similar. (b) Applying the sine law to triangles AAC,BCC and ABC,wehave

b sin(A − α) AC = , sin A a sin α b sin A sin α CC = = . sin C sin B sin C YIU : Problems in Elementary Geometry 262

It follows that sin(A − α) sin A sin α AC = b − sin A sin B sin C

AC sin(A − α) sin(C + B)sinα = − AC sin A sin B sin C =cosα − sin α[cot A +cotB +cotC] =cosα − sin α cot ω, where ω is the Brocard angle of the triangle ABC and it is well known that

cot ω =cotA +cotB +cotC.

PME769.91F.S92F.(R.S.Luthar) If ABC is a triangle in which c2 =4ab cos A cos B,prove that the triangle is isosceles.

PME773.92S.S93S.(Bankoff) In a given circle (O)achordCD is drawn to intersect diameter AOB at point E. Three circles are inscribed, the first two inthe sectors BEC and BED, ang the third in the opposite segment CED. Let the circle in sector BEC touch CE at J and let the circle in sector BED touch DE at N.Seethefigure.

Suppose the three inscribed circles have equal radii. (a) Show that CD is perpendicular to AB. AE (b) Find the ratio EB. AD (c) Find the ratio AB . CD (d) Find the ratio AB . (e) Show that the rectangle JKMN on JN as base and with opposite side KM passing through A circumscribes the third inscribed circle. (f) Show that the rectangle JKLD and NMLD are golden rectangles.

PME780.92S,(correction 92F,93S).S94F.(R.S.Luthar) Let ABCD be a parallelogram with A =60◦. Let the circle through A, B,andD intersect AC at E. See the ﬁgure. Prove that BD2 + AB · AD = AE · AC. YIU : Problems in Elementary Geometry 263

PME781.92S.S93S.(Garfunkel) Erect squares ADEF , BKL and CDGH as shown in the ﬁgure, on the segments AD, DC,andBD,whereD is any point on side CA of given triangle ABC.LetX, Y and Z be the centers of the√ erected squares. Prove that triangles ABC and XY Z are similar and the ratio of similarity is 2.

PME782.92S.S93S.(Klamkin) Bottema 12.55: for a triangle ABC with an angle ≥ 120◦, √ 2 2 2 2 2(R1 + R2 + R3) ≥ (a + b + c )+4 3, where R1, R2, R3 are the respective distances from an arbitrary point P inside the triangle to its sides. Item 12:55 further states that for a triangle in which A ≥ 120◦,

2 2 (R1 + R2 + R3) ≥ (b + c) . Show that the ﬁrst inequality is true for all triangles.

PME783.92S.S93S.(Garfunkel) In any triangle ABC, sin2 A sin A ≥ . 2 A A cos 2 cos 2 Solution. We begin by establishing the basic relations

a2 + b2 + c2 =2s2 − 2(4R + r)r, (1) ab + bc + ca = s2 +(4R + r)r, (2) abc =4Rrs. (3)

abc Since R = 4 and r = s , (3) is immediate. Also,

2 2 2 r s = = s(s − a)(s− b)(s − c) − 4 2 − = s + s bc abcs = −s4 + s2 bc − 4Rrs2, 2 2 − from which we obtain (2). (1) follows from a =( a) 2 bc. Since a2 ≥ bc with equality if and only if a = b = c,wehave,

s2 ≥ 3(4R + r)r, (4)

a with equality if and only if the triangle is equilateral. Now, since sin A = 2R ,and A s(s − a) A (s − b)(s − c) cos = , sin = , 2 bc 2 bc YIU : Problems in Elementary Geometry 264 we have sin2 A 2 A cos 2 1 2 1 2 2 a 2 a 4R 4R = s − = s − abc a(s a) abc a(s a) abc a2 r s2 = · = − 1 , 4R2s 2s2 − a2 R (4R + r)r

using (1) and (2). On the other hand, sin A A 8(s − a)(s − b)(s − c) =8 sin = A 2 abc cos 2 8(−s3 + s bc − abc) 2(−s2 + bc − 4Rr) 2r = = = . abc Rr R s2 − ≥ The inequality in question is equivalent to (4R+r)r 1 2. This is equivalent to (4) above. Equality holds if and only if the triangle is equilateral.

PME793.92F.S93S.(D.Bennewitz) Given any trapezoid, its diagonals divide its interior area into four triangular areas: A and B adjacent to the parallel bases, and C and D adjacent to the nonparallel sides. (a) Prove that the areas C and D are equal and that A · B = C · D. (b) Find area C in terms of the lengths of the altitude and the bases of the trapezoid.

PME803.93S.S94S.(R.S.Luthar) In any triangle, prove that A √ √ tan < 3 csc A. 2

PME804.93S.S94S.(R.C.Gebhardt) Show that 1 − x 4arctan = π − 4arctanx. 1+x

PME807.93S.S94S.(F.Smarandache) In terms of the lengths a, b, c of a given triangle ABC, ﬁnd the length of the segment PQ of the normal to the side BC at its midpoint M cut oﬀ by the other two sides. YIU : Problems in Elementary Geometry 265

PME808.93S.S94S.(S.H.Brown) Acircle(R) is inscribed in the unit square ABCD in the unit square ABCD and touches the sides of the square at S, T , U and V ,asshowninthe accompanying figure. Another circle (r) is inscribed in the region ASV outside circle (R)and inside the square at vertex A. (a) Find the area of the shaded region inside region ASV and outside circle (r). (b) If the sequence of smaller circles is continued indefinitely, each successive circle inscribed between the preceding and the corner A of the square, find the limit of the shaded region.

PME809.93S.S94S.(D.Iny) In triangle ABC let AD and BE be any two cevians intersect- BD AF AE BF ing at a point F .FindtheratiosDC and FD in terms of the ratios EC and FE.

PME811.93F.S94F.(T.Moore) A primitive Pythagorean triple (a, b, c)isprimeifbotha and c are prime. (a) If (a, b, c) is prime deduce that b = c − 1. (b) Find all prime, primitive Pythagorean triples in which a and c are (i) twin primes; (ii) both Mersenne primes; (iii) both Fermat primes; (iv) one a Mersenne, the other a Fermat prime.

PME812.93F.S94F.(G.P.Evanovich) If n ≥ 2 is a positive integer, prove that

n 2jπ n 2jπ cos = sin =0. n n j=1 j=1

PME813.93F.S94F.(Garfunkel) Given a triangle ABC with sides a, b, c, and a triangle 1 1 1 ≥ A B C with sides 2 (b + c), 2 (c + a), 2 (a + b). Prove that r r.

PME817.93F.S94F.(A.Cusumano) In the accompanying ﬁgure, squares CEHA and AIDB are erected externally on sides CA and AB of triangle ABC.LetBH meet IC at O and AC at G,angletCI meet AB at F . YIU : Problems in Elementary Geometry 266

(a) Prove that points D, O,andE are collinear. (b) Prove that angles HOE, EOC, AOH,andAOI are each 45◦. (c) If ACB is a right angle, then prove that E, F ,andG are collinear. Find an elegant proof for parts (a) and (b), both of which are known to be true whether the squares are rected both externally or both internally (see AMME831.49p.406–407). Part (c) is a delightful result that also should be known, but appears to be more diﬃcult to prove. (See also PME895.96F.)

PME825.94S.S95S,95F.(Bankoﬀ) Let O be a point inside the equilateral triangle ABC whosesideisoflengths.LetOA, OB, OC have lengths a, b, c respectively. Given the lengths a, b, c, ﬁnd length s. See also AMM 3904.392.S411, CMJ187.812.S824, and Bottema, On the distance of a point to the vertices of a triangle, Crux Math. 10 (1984) pp.242 – 246.

PME827.94S.S95S.(Rabinowitz) Let P be a point on diagonal BD of square ABCD and let Q be a point on side CD such that AP Q is a right angle. Prove that AP = PQ. Solution. Let the perpendicular to AB through P meet AB and CD respectively at H and π − K. Clearly, in the right triangles AP H and PKQ, AP H = 2 QP K = PQK.Furthermore, PK = DK = AH. It follows that the right triangles AP H and PKQ are congruent, and their hypotenuses AP and PQ are equal in length. Secondsolution. Note that AP QD is a cyclic quadrilateral since it has two opposite right angles. It follows that PAQ = PDQ =45◦,and in the right triangle AP Q, AP = PQ.

PME834.94S.S95S.(Klamkin) Let T and T denote two triangles with respective sides (a, b, c)and(a,b,c), where

a2 = s(s − a),b2 = s(s − b),c2 = s(s − c).

Prove that (i) s ≥ s; (ii) R ≥ R; (iii) r ≥ r; ≥ (iv) s2 s2 .

PME846.94F.S95F.(M.A.Khan) Let N, L, M be points on sides AB, BC, CA of a given triangle ABC such that AN BL CM 0 < = = = k<1. AB BC CA Let AL meet CN at P and BM at Q,andletBM and CN meet at R. Draw lines parallel to CN through A, parallel to AL through B, and parallel to BM through C.LetXY Z be the YIU : Problems in Elementary Geometry 267 triangle formed by these three new lines. Prove that (a) triangles ABC, PQR and XY Z have a common centroid; √ (v) if the areas of PQR, ABC and XY Z are in geometric progression, then k = 3 − 1.

PME847.94F.S95F(D.P.Mavlo) The midline of an isosceles trapezoid has length L and its acute angle is α. Determine the trapezoid’s area. Answer. L2 sin α.

PME851.95S.S96S. (B.Correll) In ABC let cevian AD bisect side BC and let cevians BE and BF trisect side CA.LetAD intersect BE at P and BF at R,andletCP meet BF at Q, If the area of ABC is 1, ﬁnd the area of triangle PQR.

PME852.95S.S96S. (R.H.Wu) Let E be a point inside square ABCD with BE = x, DE = y,andCE = z.Ifx2 + y2 =2z2, ﬁnd the area of ABCD in terms of x, y,z.

PME856.95S.S96S. (P.S.Bruckman) Starting with a regular n−gon whose side is of unit length, snipoﬀ congruent isosceles triangle from each of its vertices, resulting in a regular 2n−gon. Repeat the process indeﬁnitely. Find the ratio of the area of the limiting circle to that of the original n−gon. Solution. All these regular polygons have the same inscribed circle. The ratio is therefore

π π n π = π . n tan n tan n

PME865.95F.S96F.(M.A.Covas) Let ABC be a triangle with sides of lengths a, b, c, semiperimeter s and area K. Show that, if a(s − a)=4K, then the three circles centered at the vertices A, B, C and of radii s − a, s − b, s − c respectively, are all tangent to thesame straight line.

PME869.95F.S96F.(R.Behboudi) Consider an ellipse with center O and major and minor axes AB and CD respectively. Let E and F be points on segment OB so that OE2 + OF 2 = OB2.At E and F erect perpendiculars to cut arc BC at G and H respectively. Show that the areas of sectors OBH and OGC are equal.

PME871.95F.S96F.(M.A.Covas) Let ABCD be an isosceles trapezoid with major base BC. If the altitude AH is the mean proportional between the bases, then show that each side is the arithmetic mean of the bases, and show that the projection AP of the altitude on side AB is the harmonic mean of the bases. YIU : Problems in Elementary Geometry 268

PME881.96S.S97S;97F. (A.Cusumano) Let ABC be an equilateral triangle with center D.Letα be an arbitrary positive angle less than 30◦.LetBD meet CA at F .LetG be that point on segment CD such that angle CBG = α,andletE be that point on FG such that FCE = α.ProvethatDE is parallel to BC.

PME895.96F.S97F. (A.Cusumano) Let ABC be an isosceles right triangle with right angle at C. Erect squares ACEH and ABDI outwardly on side AC and hypotenuse AB.Let CI meet BH at K,andletAO meet BC at J.LetDE cut AB at F and AC at G.Itisknown (PME817.94F) that DE passes through O.LetJF meet AH at S and let JG meet BH at T . Finally, let BH and AC meet at M and let JM and CI meet at L. (a) Prove that

1. ST is parallel to DOE,

2. JK is parallel to AC,

3. JG is parallel to AB,

4. AI passes through T ,

5. JF passes through I,

6. EK passes through M,and

7. BL pases through G.

(b) Which of these results generalize to an arbitrary triangle?

PME900.96F.S97F. (H.Eves) Given the lengths of two sides of a triangle and that the medians to those two sides are perpendicular to each other, construct the triangle with euclidean tools.

PME910.97S. (W. Chau) A triangle whose sides have lengths a, b, c has area 1. Find the line segment of minimum length that joins two sides and separates the interior of the triangle into two parts of area α and 1 − α,whereα is a given number between 0 and 1.

PME910.97S.S98S. (N.Schaumberger) If a, b,andc are the lengths of the sides of a triangle with semiperimeter s and area K, show that s a/(s−a) s b/(s−b) s c/(s−c) s4 + + ≥ . s − a s − b s − c K2 YIU : Problems in Elementary Geometry 269

PME919.97F.S98F. (C.W.Dodge) Erect directly similar nondegenerate triangles DBC, ECA, FAB on sides BC, CA, AB of triangle ABC.AtD, E, F center circles of radii k · BC, k · CA, k · AB respectively for ﬁxed positive k.LetP be the radical center of the three circles. If P lies on the Euler line of the triangle, show that it always falls on the same special point.

PME924.97F.S98F. (G.Tsapakidis) Find an interior point of a triangle so that its projections on the sides of the triangle are the vertices of an equilateral triangle. Comment: The solution given by W.H.Peirce can actually be adapted to give a simple description of the points. Peirce solved the problem by calculating the barycentric coordinates of the point P . Homogenizing, we obtain

a sin(α ± 60◦):b sin(β ± 60◦):c sin(γ ± 60◦)

√ These points divide the segment OK harmonically, in the ratio a2 + b2 + c2 :4 3. They are the isogonal conjugates of the points a b c : : , sin(α ± 60◦) sin(β ± 60◦) sin(γ ± 60◦) which are the isogonal centers of the triangle.

PME936.98S. (J.Garfunkel) Given the Malfatti conﬁguration, where three mutually external, mutually tangent circles with centers A, B, C are inscribed in a triangle ABC so that circle (A) is tangent to the two sides of angle A,circle(B) is tangent to the sides of angle B,and(C) to the sides of C.If A ≤ B ≤ C,and A< C, then prove that we have C − A < C − A.

PME937.98S. (R.S.Luthar) Let I be the incenter of triangle ABC,letAI cut the triangle’s circumcircle again at point D,andletF be the foot of the perpendicular dropped from D to side BC, as shown in the ﬁgure. Prove that DI2 =2R · DF,whereR is the circumradius of triangle ABC.

PME938.98S. (R.S.Luthar) Find the locus of the midpoints M of the line segment in the ﬁrst quadrant lying between the two axes and tangent to the unit circle centered at the origin.

PME939.98S. (Khiem Viet Ngo) In the accompanying ﬁgure both quadrilaterals ABCD and MNPQ are squares, each side of square ABC has length 1, and the ﬁve inscribed circle are all congruent to one another. Find their common radius. YIU : Problems in Elementary Geometry 270

PME945.98F. (J.Garfunkel) Let A, B, C be the angles of a triangle and A, B, C those of another triangle with A ≥ B ≥ C, A>C, A ≥ B ≥ C,andA >C. Prove or disprove that if A − C ≥ 3(A − C), then

A cos ≤ sin A. 2

PME946.98F. (Ayoub B. Ayoub) Let M be a point inside (outside) triangle ABC if A is acute (obtuse) and let MBA+ MCA =90◦. (a) Prove that (BC · AM)2 =(AB · CM)2 +(CA · BM)2, (b) Show that the Pythagorean theorem is a special case of the formula of part (a).

PME950.98F. (S.B.Karmakar) Let c>b>a>0 be the lengths of the sides of an obtuse m triangle; let m be a prime and n an evern positive integer such that 1

ad + bd = cd cannot be satisﬁed if a, b,andc are relatively prime in pairs.

PME952.98F. (P.A.Lindstrom) Let A, B, C denote the measures of the angles and a, b, c the opposite sides of a triangle. Show that

(a + b + c)(b + c − a)(c + a − b)(a + b − c)(bc + ca + ab) sin A sin B+sinB sin C+sinC sin A = . 4a2b2c2 YIU : Problems in Elementary Geometry 271 Crux Mathematicorum Geometry Problems (1975 – 2000)

Crux 5.S1.15.(L.Sauv´e) Prove that, if (a, b, c)and(a,b,c) are primitive Pythagorean triples, with a>b>cand a >b >c, then either

aa ± (bc − cb)oraa ± (bb − cc) are perfect squares.

Crux 14.S1.28. (V.Linis) If a, b, c are the lengths of three segments which can form a 1 1 1 triangle, show the same for a+c , b+c , a+b .

Crux 15.S1.28. (H.G.Dworkschak) Let A, B, C be three distinct points on a rectangular hyperbola. Prove that the orthocenter of ABC lies on the hyperbola. k k k [Solution by L´eo Sauv´e]: If A, B, C are the points (a, a ), (b, b ), (c, c ) on the hyperbola k − k2 xy = k, the orthocenter is the point (t, t ), with t = abc .

Crux 18.S1.31;2.42,69. (J.Marion) Montrer que, dans un triangle rectangle dont les cˆote ont 3, 4 et 5 unit´es de longueur, aucun des angles aigus n’est un multiple rationnel de π.

Crux 19.S1.32. (H.G.Dworkschak) How many diﬀerent triangles can be formed from n straight rods of lengths 1,2, . . . , n ? See also CMJ503.933.

Crux 24.S1.42. (V.Linis) A paper triangle has base 6 cm and height 2 cm. Show that by three or fewer cuts the sides can cover a cube of edge 1 cm.

Crux 27.S1.44. (L.Sauvé) Soient A, B,etC les angles d’un triangle. Il est facile de vérifer que si A = B =45◦,alors cos A cos B +sinA sin B sin C =1. La proposition réciproque est-elle vraie ?

Crux 29.S1.45. (V.Linis) Cut a square into a minimal number of triangles with all angles acute. YIU : Problems in Elementary Geometry 272

Crux 32.S1.59. (V.Linis) Construct a square given a vertex and a midpoint of one side.

Crux 33.S1.60. (V.Linis) On the sides CA and CB of an isosceles right triangle ABC, points D and E are chosen such that CD = CE. The perpendiculars from D and C on AE intersect the hypotenuse AB in K and L respectively. Prove that KL = LB.

Crux 37.S1.62*. (M.Poirier) E, F , G,andH are the midpoints of the sides AB, BC, CD and DA respectively of the convex quadrilateral ABCD. EX, FY, GZ and HT are drawn 1 1 externally perpendicular to AB, BC, CD and DA, respectively, and EX = 2 AB, FY = 2 BC, 1 1 ⊥ GZ = 2 CD,andHT = 2 DA.ProvethatXZ = YT and XZ YT. [Restatement]: If squares on constructed externally on the sides of a convex quadrilateral, the centers of the squares form a quadrilateral whose diagonals are equal and perpendicular to each other. [Editor’s Comment]: This theorem is due to H. van Aubel, who was a professor in the athénée of Antwerp around 1880. It appears as Proble 10 on p.23 of Coxeter’s Introduction to Geometry, and a solution (different from those given in Crux) is given at the back of the book. Paul J. Kelly [von Aubel’s quadrilateral theorem, Math. Mag. 39 (1966) 35] generalized it to four non-coplanar points.

Crux 38.S1.63*. (L.Sauv´e) Consider the two triangles ABC and PQR show below. In ABC, ADB = BDC = CDA = 120◦.ProvethatX = u + v + w.

Crux 39.S1.64*;2.7. (M.Poirier) On donne un point P à l’intérieur dún triangle équilatéral ABC tel que les longueurs des segments PA, PB, PC sont 3,4, et 5 unités respectivment. Calculer láire du ABC.

Crux 42.S1.73. (V.Linis) Find the area of quadrilateral as a function of its four sides, YIU : Problems in Elementary Geometry 273 given that the sums of opposite angles are equal.

Crux 44.S1.74. (V.Linis) Construct a square ABCD given its center and any two points M and N on its two sides BC and CD respectively.

Crux 46.S1.75. (F.G.B.Maskell) 1 1 1 1 + + = . ha hb hb r

Solution. 2 = aha = bhb = chc =(a + b + c)r.

Crux 56.S1.89. (F.G.B.Maskell) What is the area of a triangle in terms of its medians ?

Crux 62.S1.99. (F.G.B.Maskell) Prove that if two circle touch externally, their common tangent is a mean proportion between their diameter.

Crux 63.S1.99. (H.G.Dworkschak) From the centers of each of two nonintersecting circles tangents are drawn to the other circle. Prove that the chords PQ and RS are equal in length. (I have been told that this problem originated with Newton, but have not been able to ﬁnd the exact reference).

Crux 67.S1.101. (V.Linis) Show that in any convex 2n−gon there is a diagonal which is not parallel to any of its sides.

Crux 70.S1.102. (V.Linis) Show that for any 13−gon there exists a straight line containing only one of its sides. Show also that for every n>13 there exists an n−gon for which the above statement does not hold. YIU : Problems in Elementary Geometry 274

Crux 73.S2.9. (V.Linis) Is there a polyhedron with exactly ten pentagons as faces ?

Crux 74.S2.10. (V.Linis) Prove that if the sides a, b, c of a triangle satisfy a2 + b2 = kc2, 1 then k>2 .

Crux 75.S2.10*. (R.D.Butterill) M is the midpoint of chord AB of the circle with center C shown in the ﬁgure below. Prove that RS > MN. See also Crux 110.

MN [Solution:] RS =cosNPS.

Crux 86.S2.30. (V.Linis) Find all rational Pythagoras triples (a, b, c) such that

a2 + b2 = c2, and a + b = c2.

Crux 89.S2.33. (V.Bradley and C.Robsertson) A goat is tethered to a point on the circumference of a circular ﬁeld of radius r by a rope of length . For what value of will it be able to graze over exactly half of the ﬁeld ?

Crux 93.S2.45,111. (H.G.Dworkschak) Is there a convex polyhedron having exactly seven edges ? See also Crux 121.

Crux 94.S2.46. (H.G.Dworkschak) If, in a tetrahedron, two pairs of opposite edges are orthogonal, is the third pair of opposite necessarily orthogonal ?

Crux 96.S2.48. (V.Linis) By euclidean methods divide a 13◦ angle into 13 equal parts. YIU : Problems in Elementary Geometry 275

Crux 102.S2.73. (L.Sauv´e) Si, dans un ABC,ona =4,b =5,etc =6,montrerque C =2A.

cos α sin α − Crux 103.S2.74. (Dworschak) If cos β + sin β = 1, prove that

cos3 β sin3 β + =1. cos α sin α [Solution by L´eo Sauv´e]: The following are equivalent:

cos α sin α − 1. cos β + sin β = 1;

cos3 β sin3 β 2. cos α + sin α =1; − 1 3. sin(α + β)= 2 sin 2β.

This last item is equivalent to

sin(β + β)+sin(β + α)+sin(α + β)=0.

By Crux 132, this means that the normal to the ellipse

x2 y2 + =1 a2 b2 at points with eccentric angles α, β, β are concurrent. This is the case if and only if the center of curvature for β,namely, c2 c2 cos3 β, − sin3 β a b lies on the normal at α. ...

Crux 106.S2.78. (V.Linis) Prove that, for any quadrilateral with sides a, b, c, d, 1 a2 + b2 + c2 > d2. 3

4m Crux 107.S2.79. (V. Linis) For which integers m and n is the ratio 2m+2n−mn an integer ? This problem has a geometric application.

Crux 109.S2.81. (L. Sauv´e) (a) Prove that rational points are dense on any circle with rational center and rational radius. YIU : Problems in Elementary Geometry 276

(b) Prove that if the radius is rational the circle may have inﬁnitely many rational points. (c) Prove that if even one coordinate of the center is irrational, the circle has at most two ratinal points.

Crux 110.S2.84. (Dworschak) (a) Let AB and PR be two chords of a circle intersecting at Q.IfA, B,andP are kept ﬁxed, characterize geometrically the position of R for which the length of QR is maximal.

(b) Give a euclidean construction for the point R which maximizes the length of QR,or show that no such construction is possible. [Solution.] (a) QR bisected by the diameter perpendicular to AB. (b) Not constructible in general.

Crux 113.S2.97. (L. Sauvé) Si u =(b, c, a)etv =(c, a, b) sont deux vecteurs non nuls dans l’espace euclidien réel à trois dimensions, quelle est la valeur maximale de l’angle (u, v) entre u et v ? Quand cette valeur maximale est-elle atteinte ?

Crux 115.S2.98,111,137*. (V. Linis) Prove the following inequality of Huygens: π 2sinα + tanα ≥ 3α, 0 ≤ α< . 2 See also Crux 167.

Crux 119.S2.102. (J.A.Tierney) A line through the ﬁrst quadratnt point (a, b)formsa right triangle with the positive coordinate axes. Find analytically the minimum perimeter of the triangle.

Crux 120.S2.102,139. (J.A.Tierney) Given a point P inside an arbitrary angle, give a euclidean construction of the line through P that determines with the sides of the angle a triangle YIU : Problems in Elementary Geometry 277

(a) of minimum area; (b) of minimum perimeter.

Crux 121.S2.121,139. (L. Sauv´e) For which n is there a convex polyhedron having exactly n edges ? See also Crux 93.

Crux 125.S2.120.* (B. Vanbrugghe) A l’aide d’un compas seulement, d´eterminer le centre inconnu d’un cercle donn´e.

Crux 126.S2.123. (V. Linis) Show that, for any triangle ABC,

|OA|2 sin A + |OB|2 sin B + |OC|2 sin C =2.

Crux 127.S2.124,140,221. (V. Linis) A, B, C, D are four distinct points on a line. Construct a square by drawing two pairs of parallel lines through the four points.

Crux 132.S2.142,172;3.11. (L. Sauv´e) If cos θ =0,andsin θ =0for θ = α, β, γ,prove that the normals to the ellipse x2 y2 + =1 a2 b2 at the ponts of eccentric angles α, β, γ are concurrent if and only if

sin(α + β)+sin(β + γ)+sin(γ + α)=0.

Crux 134.S2.151,173,222;3.12,44. (K.S.Williams) ABC is an isosceles triangle with ABC = ACB =80◦. P is the point on AB such that PCB =70◦. Q is the point on AC such that QBC =60◦.Find PQA. See also Crux 175.

Crux 136.S2.153. (S.R.Conrad) In ABC, C is on AB such that AC : CB =1:2and B is on AC such that AB : BC =4:3.LetP be the intersection of BB and CC,andletA be the intersection of BC and ray AP .FindAP : PA.

Crux 137.S2.156. (V. Linis) On a rectangular billiard table ABCD,whereAB = a and BC = b, one ball is at a distance p from AB and at a distance q from BC, and another ball is at the center of the table. Under what angle α (from AB) must the ﬁrst ball be hit so that after the rebounds from AD, DC, CB it will hit the other ball ? YIU : Problems in Elementary Geometry 278

Crux 139.S2.158. (D.Pedoe) ABCD is a parallelogram, and a circle γ touches AB and BC and intersects AC in the points E and F . Then there exists a circle δ which passes through E and F and touches AD and DC. Prove this theorem without using Rennie’s lemma. See Crux 2. p.65.

Crux 140.S3.13,46. (D.Pedoe) (The Veness Problem) A paper cone is cut along a generator and unfolded into a plane sheet of paper. what curve in the plane do the originally plane sections of the cone become ?

Crux 141.S2.174. (Bankoﬀ) What is wrong with the following proof of the Steiner – Lehmus theorem ? At the midpoints of the angle bisectors, I erect two perpendiculars which meet in O;withO as center and AO as radius, I describe a circle which will evidently pass through the points A, M, N, C. MN Now, the angles MAN, MCN are equal since the measure of each is arc 2 ; hence BAC = ACB, and triangle ABC is isosceles.

Crux 144.S2.180. (V.Linis) In a triangle ABC, the medians AM and BN intersect at G. If the radii of the inscribed circles in triangle ANG and BMG are equal, show that ABC is an isosceles triangle.

Crux 147.S2.183. (S.R.Conrad) In square ABCD, AC and BD meet at E.PointF is in CD and CAF = FAD.IfAF meets ED at G and if EG = 24, ﬁnd CF.

Crux 148.S2.183. (S.R.Conrad) In ABC, C =60◦,and A is greater than B.The bisector of C meets AB in E.IfCE is a mean proportional between AE and EB, ﬁnd B.

Crux 155.S2.198;3.22. (S.R.Conrad and I.Ewen) A plane is tessellated by regular hexagons when the plane is the union of congruent regular hexagonal closed regions which have disjoint interiors. A lattice point of this tessellaton is any vertex of any of the hexagons. Prove that no four lattice points of a regular hexagonal tessellation of a plane can be the vertices of a regular 4−gon.

Crux 158.S2.201. (A.Bourbeau) Devise a euclidean construction to divide a given line segment into two parts such that the sum of the squares on the whole segment and on one of its parts is equal to twice the square on the other part. Solution. Let AB be a given segment. Extend AB to a point B so that BB = AB. Construct equilateral triangle ABC.OnAB mark a point P such that BP = BC. Then the sum of the squares on AB and AP is twice the square on PB. YIU : Problems in Elementary Geometry 279

Remark. This is the same as Trigg’s solution. How would Euclid have justifed such a construction?

Crux 165.S2.230. (D. Eustice) Prove that, for each choice of n points in the plane (at least two distinct), there exists a point on the unit circle such that the product of the distances from the point to the chosen points is greater than one. See also Crux 173.

Crux 167.S3.23. (L. Sauv´e) The ﬁrst half of the Snellius - Huygens double inequality

1 3sinα π (2 sin α +tanα) >α> , 0 <α< , 3 2+cosα 2 was proved in Crux 115. Prove the second half in a way that could have been understood before the invention of calculus. See also Crux 115.

Crux 168.S2.233. (Garfunkel) Let ta, tb, tc be the lengths of the bisectors of a triangle, and Ta, Tb, Tc these angle bisectors extended until they are chords of the circumcircle. Prove that abc = tatbtcTaTbTc.

Crux 171.S3.26. (D.Sokolowsky) Let P1 and P2 denote, respectively, the perimeters of ABE and ACD as shown. Without using circles, prove that

P1 = P2 ⇒ AB + BF = AD + DF.

See also Crux 2, p.108. YIU : Problems in Elementary Geometry 280

Crux 173.S3.47,68. (D. Eustice) For each choice of n points on the unit circle (n ≥ 2), there exists a point on the unit circle such that the product of the distances to th chosen points is greater than 2. Moreover, the product is 2 if and only if the n points are the vertices of a regular polygon. See also Crux 165.

Crux 175.S3.49. (A. Dunkels) Consider the isosceles triangle ABC with vertical angle A =20◦.OnAC, one of the equal sides, a point D is marked oﬀ so that |AD| = |BC| = b.Find the measure of ABD. See also Crux 134.

Crux 177.S3.50*,132. (K.S.Williams) P is apoint on th diameter AB of a circle whose center is C.OnAP , BP as diameters, circles are drawn. Q is the center of a circle which touches these three circles. Whati s the loocus of Q as P varies ?

Crux 180.S3.50. (K.S.Williams) Through O, the midpoint of a chord AB of an ellipse, is drawn any chord POQ. The tangents to the ellipse at P and Q meet AB at S and T respectively. Prove that AS = BT.

Crux 181.S3.51. (Trigg) A polyhedron has one square face, two equaliteral triangular faces attached to opposite sides of the square, and two isosceles trapezoidal faces, each with one edge equal to twice a side of the square. What is the volume of this pentahedron in terms of a side of the square ?

Crux 189.S3.74,193,252. (K.S.Williams) If a quadrilateral circumscribes an ellipse, prove that the line through the midpoints of its diagonals passes through the center of the ellipse.

Crux 192.S3.79*. (Honsberger) Let D, E, F denote the feet of the altitudes of ABC, and let (X1,X2), (Y1,Y2), (Z1,Z2) denote the feet of perpendiculars from D, E, F respectively, upon the other two sides of the triangle. Prove that the 6 points X1, X2, Y1, Y2, Z1, Z2 lie on a circle.

Crux 199.S3.112,298. (Dworschak) If a quadrilateral is circumscribed about a circle, prove that its diagonals and the two chords joining the points of contact of opposite sides are all concurrent.

Crux 200.S3.134,228. (L.Sauv´e) (a) Prove that there exist triangles which cannot be dissected into two or three isosceles triangles. (b) Prove or disprove that, for n ≥ 4, every triangle can be dissected into n isosceles triangles. YIU : Problems in Elementary Geometry 281

Crux 206.S3.143. (S.R.Conrad) A circle intersects the sides BC, CA and AB of triangle ABC in the pairs of points X, X, Y , Y ,andZ, Z respectively. If the perpendiculars at X, Y and Z to the respective sides BC, CA and AB are concurrent at a point P , prove that the respective perpendiculars at X, Y and Z to the sides BC, CA and AB are concurrent at a point P .

Crux 210.S3.160,196. (Klamkin) P , Q, R denote points on the sides BC, CA and AB respectively of a given triangle ABC. Determine all triangles ABC such that if BP CQ AR 1 = = = k(=0 , , 1), BC CA AB 2 then PQR (in some order) is similar to ABC.

Crux 213.S3.160. (W.J.Blundon) (a) Prove that the sides of a triangles are in arithmetric progression if and only if s2 =18Rr − 9r2. (b) Find the correpsonding result for geometric progression.

Crux 218.S3.172. (G.W.Kessler) Everyone knows that the altitude to the hypotenuse of a right triangle is the mean proportional between the segments of the hypotenuse. The median to the hypotenuse also has this property. Does any other segment from vertex to hypotenuse have the property ?

Crux 220.S3.175. (D.Sokolowsky) C is a point on the diameter AB of a circle. A chord through C, perpendicular to AB, meets the circle at D. A chord through B meets CD at T and arc AD at U. Prove that there is a circle tangent to CD at T and to arc AD at U.

Crux 222.S3.200. (B.McColl) Prove that π 2π 3π 4π 5π √ tan tan tan tan tan = 11. 11 11 11 11 11

Crux 223.S3.202. (S.R.Conrad) Without using any table which lists Pythagorean triples, ﬁnd the smallest integer which can represent the area of two noncongruent primitive Pythagorean triangles.

Crux 224.S3.203. (Klamkin) Let P be an interior point of a given n−dimensional simplex of vertices A1, A2, ..., An+1.LetPi,(i =1, 2,...,n + 1) denote points on AiP such that AiPi 1 PiP = ni . Finally, let Vi denote the volume of the simplex cut oﬀ from the given simplex by a hyperplane through Pi parallel to the face of the given simplex opposite Ai. Determine the minimum value of Vi and the location of the corresponding point P . YIU : Problems in Elementary Geometry 282

Crux 225.S3.204. (D.Sokolowsky) C is a point on the diameter AB of a circle. A chord through C, perpendicular to AB, meets the circle at D. Two chords through B meets CD at T1, T2,andarcAD at U1,.U2 respectively. It is known from Problem 220 that there are circles C1, C2 tangent to CD at T1, T2 and to arc AD at U1, U2 respectively. Prove that the radical axis of C1 and C2 passes through B.

Crux 229.S3.231. (K.M.Wilke) On an examinantion, one question asked for the largest angle of the triangle with sides 21, 41, 50. A student obtained the correct answer as follows: Let C denote the desired angle; then 50 9 sin C = =1+ . 51 41 ◦ 9 ◦ But sin 90 =1and 41 =sin12 40 49 .Thus, C =90◦ +12◦4049 = 102◦4049, which is correct. Find the triangle of least area having integer sides and possessing this property.

Crux 232.S3.238;4.17. (V.Linis) Given are five points A, B, C, D, E in the plane, together with the segments joining all pairs of distinct points. The areas of the five triangles BCD, EAB, ABC, CDE, DEA being known, find the area of the pentagon ABCDE. The above problem with a solution by Gauss was reported by Schumacher. The problem was given by Möbius in his book on the Observatory of Leipzig, and Gauss wrote his solution in the margins of the book.

Crux 233.S3.252. (V.Linis) The three points (1), (2), (3) lie in this order on an axis, and the distances [1, 2] = a,and[2, 3] = b are given. Points (4) and (5) lie on one side of the axis, and the distance [4, 5] = 2c>0 and the angles (415) = v1, (425) = v2, (435) = v3 are also known. Determine the position of the points (1), (2), (3) relative to (4) and (5). Gauss gave a solution to this problem which was found in a boook on navigation [Handbuch der Schiﬀahrtskunde von C. R´umker, 1850, p.76].

2nπ ± π Crux 234.(correction 3.154).S3.257. (V.Linis) If sin 13 = sin 13 ,provethat π 2π 4π 2n−1π 1 cos cos cos ···cos = ± . 13 13 13 13 2n Gauss’ remark: Inspect a polygon !

Crux 242.S3.266*. (B.McColl) Give a geometrial construction for determining the focus of a parabola when two tangents and their points of contact are given. YIU : Problems in Elementary Geometry 283

Crux 244.S4.19*. (S.R.Conrad) Solve the following problem, which can be found in IntegratedAlgebra andTrigonometry , by Fisher and Ziebur, Prentice Hall, (1957) p. 259: A rectangular stripof carpet3 ft. wide is laid diagonally across the ﬂoor of a room 9 ft. by 12 ft. so that each ofthe four corners of the striptouches a wall. How long is the strip?

Crux 245.S4.21. (Trigg) Find the volume of a regular tetrahedron in terms of its bimedian b. (A bimedian is a segment joining the midpoints of opposite edges).

Crux 248.(correction 3.154).S4.26 (D.Sokolowsky) Circle (Q) is tangent to circles (O), (M), N) as shown in the ﬁgure, and FG is the diameter of (Q) parallel to diameter AB of (O). W is the radical center of circles (M), (N), (Q). Prove that WQ is equal to the circumradius of PFG. YIU : Problems in Elementary Geometry 284

Crux 255.S4.52. (B.Hornstein) In the adjoining ﬁgure, the measures of certain angles are given. Calculate x in terms of α, β, γ, δ.

Crux 256.S4.53,102. (H.L.Nelson) Prove that an equilateral triangle can be dissected into ﬁve isosceles triangles, n of which are equilateral, if and only if 0 ≤ n ≤ 2.

Crux 257.S4.54*. (W.A.McWorter) Can one draw a line joining two distant points with a BankAmericard ?

Crux 260.S4.58;8.80. (W.J.Blundon) Given any triangle (other than equilateral), let P represent the projection of the incenter I on the Euler line OGNH.ProvethatP lies between G and H. In particular, prove that P coincides with N if and only if one angle of the given triangle is 60◦.

Crux 268.S4.78*. (G.Salvatore) Show that in ABC with a ≥ b ≥ c, the sides are in arithmetic progression if and only if B C A 2cot = 3(tan +tan ). 2 2 2

Crux 270.S4.82*. (D.Sokolowsky) Call a chord of a triangle a segment with enpoints on the sides. Show that for very acute angled triangle there is a unique point P through which pass three equal chords each of which is bisected by P . See also editor’s comment on Crux 624.S8.111.

Crux 271.S4.84. (S.Avital) Find all possible triangle ABC which have the property that one can draw a line AD, outside the triangular region, on the same side of AC as AB,which meets CB extended in D so that triangles ABD and ACD will be isosceles. YIU : Problems in Elementary Geometry 285

Crux 275.S4.105. (G.W.Kessler) Given are the points P (a, b)andQ(c, d), where a, b, c, d are all rational. Find a formula for the number of lattice points on the segment PQ.

Crux 278.S4.110. (W.A.McWorter) If each of the medians of a triangle is extended 4 beyond the sides of the triangle to 3 its length, show that the three new points formed and the vertices of the triangle all lie on an ellipse.

Crux 279.S4.110. (F.G.B.Maskell) On donne sur une droite trois points distincts A, O, B tels que O est entre A et B,etAO = OB. Montrer que les trois coniques ayant deux foyers et un sommet aux trois points donn´es sont concourantes en deux points.

Crux 284.S4.115. (W.A.McWorter) Given a sector AOD of a circle (see ﬁgure), can a straightedge and compass construct a line OB so that AB = AC ?

Crux 288.S4.136*. (W.J.Blundon) Show how to construct a triangle given the circumcenter, the incenter and one vertex.

Crux 292.S4.148*. (Trigg) Fold a square piece of paper to form four creases that determine angles with tangents of 1, 2, and 3.

Crux 309.S4.200. (Peter Shor) Let ABC be a triangle with a ≥ b ≥ c or a ≤ b ≤ c.Let D and E be the midpoints of AB and AC, and let the bisectors of angles BAE and BCD meet at R.Provethat (a) AR ⊥ CR if and only if 2b2 = c2 + a2; (b) if 2b2 = a2 + c2,thenR lies on the median from B. Is the converse of b true? See Crux 210.S4.13.

Crux 313.S4.207*. (Leon Bankoﬀ) In an RMS triangle ABC (that is, a triangle in which YIU : Problems in Elementary Geometry 286

2b2 = c2 + a2), prove that GK, the join of the centroid and the symmedian point, is parallel to the base b.

Crux 315.S4.227. (O.Ramos) Prove that if two points are conjugate with respect to a circle, the sum of their powers is equal to the square of the distance between them.

Crux 317.S4.230*. (J.G.Propp) In triangle ABC,letD and E be the trisection points of side BC with D between B and E,letF be the midpoint of side AC,andletG be the midpoint of side AB.LetH be the intersection of segments EG and DF.FindtheratioEH : HG by means of mass points or otherwise.

Crux 318.S4.231. (C.A.Davis) Given any triangle ABC, thinking of it as in the complex plane, two ponts L and N may be deﬁned as the stationary values of a cubic that vanishes at the vertices A, B, C.ProvethatL and N are the foci of the ellipse that touches the sides of the triangle at their midpoints, which is the inscribed ellipse of maximal area. See also Crux 659.

Crux 320.S4.238*. (D.Sokolowsky) The sides of triangle ABC are trisected by the points P1, P2, Q1, Q2, R1, R2 as shown in the ﬁgure below. Show that (a) P1Q1R1 ≡P2Q2R2; | | 1 | | (b) P1Q1R1 = 3 ABC , (c) the sides of triangles P1Q1R1 and P2Q2R2 trisect each other; (d) if M1 is the midpoint of AB,thenC, S, T , M1 are collinear.

Crux 322.S4.254. (H.Sitomer) In parallelogram ABCD,angleA is acute and AB =5. YIU : Problems in Elementary Geometry 287

Point E is on AD with AE =4andBE = 3. A line through B, perpendicular to CD, intersects CD at F .IfBF = 5, ﬁnd EF. A geometric solution (no trigonometry) is desired.

Crux 325.S4.258*;5.49. (B.C.Rennie) It is well known that if you put two pins in a drawing board and a loop of string around them you can draw an ellipse by pulling the string tight with a pencil. Now suppose that instead of two pins you use an ellipse cut out from plywood. Will the pencil in the loop of string trace out another ellipse?

Crux 330.S4.263*. (M.S.Klamkin) It is known that if any one of the following three conditions holds for a given tegrahedron, then the four faces of the tetrahedron are mutually congruent, i.e., the tetrahedron is isosceles: (1) The perimeter of the four faces are mutually equal. (2) The areas of the four faces are mutually equal. (3) the circumcircles of the four faces are mutually congruent. Does the condition that the incircles of the four faces be mutually congruent, also, imply that the tetrahedron be isosceles? See also Crux 478.S6.217*.

Crux 338.S4.290*. (W.A.McWorter) Can one locate the center of a circle with a VISA card?

Crux 353.S5.56*. (O.Ramos) Prove that if a triangle is self polar with respect to a parabola, its nine - point circle passes through the focus.

Crux 363.S5.110*. (R.H.Eddy) The following generalization of the Fermat point is known: if similar isosceles triangles BCA, CAB, ABC are constructed externally to triangle ABC, then AA, BB, CC are concurrent. Determine a situation in which AA, BB, CC are concurrent if the constructed triangles are isosceles but not similar.

i Crux 364.S5.113*. (S.R.Mandan) In the euclidean plane, if x1,(x = a, b; i =0, 1, 2), 1 are the 2 triads of perpendiculars to a line p from 2 triads of points Xi (X = A, B)onp and i i (X) a pair of triangles with vertices Xi on x − 1 and sides x opposite Xi such that the three i 1 perpendiculars to b from Ai concur at a point G, then it is true for every member of the 3- 1 i parameter family f(B) of triangles like (B); andthe3perpendicularsfromBi to the sides a of any memebr of the 3-parameter family f(A) of triangles like A concur at a point G if and only if 1 1 1 1 A0A1 B0 B1 1 1 = 1 1 . A1A2 B1 B2 YIU : Problems in Elementary Geometry 288

Crux 365.S5.114*. (K.Satyanarayana) A scalene triangle ABC is such that the external bisectors of angles B and C are of equal length. Given the lengths of sides b, c,(b>c), ﬁnd the length of the third side a and show that its value is unique.

Crux 379.S5.149*. (P.Arends) Construct a triangle ABC, given angle A and the lengths of side a and ta (the internal bisector of angle A). Editor’s Remark: This can be found in Casey, Sequel, p.80. It also appeared in the Monthly, in 1906, 1931, and 1974. See AMM E2499.

Crux 383.S5.174*. (D.Skolowsky) Let ma, mb, mc be the medians of triangle ABC. Prove that (a) if ma : mb : mc = a : b : c, then triangle ABC is equilateral; (b) if mb : mc = c : b, then either (i) b = c or (ii) quadrilateral AEGF is cyclic; (c) if both (i) and (ii) hold in (b), then triangle ABC is equilateral.

Crux 386.S5.179. (Francine Bankoﬀ) AsquarePQRS is inscribed in a semicircle (O) with PQfalling along diameter AB. A right triangle ABC, equivlaent to the square, is inscribed in the same semicircle with C lying on the arc RB. Show that the incenter I of triangle ABC lies at the intersection of SB and RQ,andthat √ RI SI 1+ 5 = = , the golden ratio. IQ IB 2

Crux 388.S5.201. (W.J.Blundon) Prove that the line containing the circumcenter and the incenter of a triangle is parallel to a side of the triangle if and only if (2R − r)2(R + r) s2 = . R − r YIU : Problems in Elementary Geometry 289

Crux 397.S5.234. (J.Garfunkel) Given is triangle ABC with incenter I.LinesAI, BI, √CI aredrawntomeettheincircle(I)fortheﬁrsttimeinD, E, F respectively. Prove that 3(AD + BE + CF) is not less than the perimeter of the triangle of maximum perimeter that can be inscribed in circle (I).

Crux 412.S5.300* (K.Satyanarayana) The sides BC, CA, AB of triangle ABC are produced respectively to D, E, F so that CD = AE = BF. Show that triangle ABC is equilateral if (and only if) DEF is equilateral.

Crux 414.S5.304*. (B.C.Rennie) A few years ago a distinguished mathematician wrote a book saying that the theorems of Ceva and Menelaus were dual to each other. Another distinguished mathematician reviewing the book wrote that they were not dual. Explain why they were both right, or if you are feeling in a sour mood, why they were both wrong.

Crux 415.S5.306*. (A.Liu) Is there a euclidean construction of a triangle given two sides and the radius of the incircle?

Crux 416.S5.307. (W.A.McWorter) Let A0BC be a triangle aand a a positive number les than 1. Construct P1 on A0B so hat A0P1/A0B = a.ConstructA1 on P1C so that P1A1/P1C = a. Inductively construct Pn+1 on AnB so that AnPn+1/AnB = a and construct An+1 on Pn+1C so that Pn+1An+1/Pn+1C = a. Show htat lla the Pi areonalineandallthe Ai are on a line, the two lines being parallel.

Crux 419.S6.19. (G.Ramanaiah) A variable point P describeds the ellipse

x2 y2 + =1. a2 b2 Does it make sense to speak of “the mean distance of P from a focus S” ? If so, what is this mean distance?

Crux 420.S6.21*. (J.A.Spencer) Given an angle AOB, ﬁnd an economical euclidean construction that will quadrisect the angle. Proposer: 5 euclidean operations suﬃce.

Crux 422.S6.24*. (Pedoe) The line and m are parallel edges of a strip of paper and P1, Q1 are points on and m respectively. Fold P1Q1 along and crease, obtaining P1Q2 as the crease. Fold P1Q2 along m and crease, obtaining P2Q2.FoldP2Q2 along and crease, obtaining P2Q3. If the process is continued indeﬁnitely, show that the triangle PnPn+1Qn+1 tends towards an equilateral triangle. YIU : Problems in Elementary Geometry 290

Crux 423.S6.26*. (J.Garfunkel) A B − C t ≤ cos2 cos ≤ m . a 2 2 a

Crux 428.S6.50. (J.A.Spencer) Let AOB be a right - angled triangle with legs OA =2OB. Use it to ﬁnd an economical euclidean construction of a regular pentagon whose side is not equal to any side of AOB. “Economical” means here using the smallest possible number of euclidean operations: setting a compass, striking an arc, drawing a line.

Crux 435.S6.60. (J.A.H.Hunter) In rectangle ABDF , AC = 125, CD = 112, DE = 52, as shown in the ﬁgure,and AB, AD, AF are also integral. Evaluate EF.

Crux 444.S6.90*. (D.Sokolowsky) A circle is inscribed in a square ABCD. A second circle on diameter BE touches the ﬁrst circle. Show that AB =4BE.

Crux 445.S6.92. (Jordi Dou) Consider a family of parabolas escribed to a given triangle. YIU : Problems in Elementary Geometry 291

To each parabola corresponds a focus F and a point S of intersection of the lines joining the vertices of the triangle to the points of contact with the opposite sides. Prove that all lines FS are concurrent.

Crux 450.S6.120*;214. (Andy Liu) Triangle ABC has a ﬁxed base BC and a ﬁxed inradius. Describe the locus of A as the incircle rolls along BC.WhenisAB of minimal length (geometric characterization desired)?

Crux 454.S6.125*. (R.R.Tiwari) (a) Is there a euclidean construction for a triangle ABC given the lengths of its internal angle bisectors ta, tb, tc ? (b) Find formulas for the sides a, b, c in terms of ta, tb, tc. See also Crux 749. See also Crux 749.

Crux 456.S6.128. (O.Ramos) Let ABC be a triangle and P any point in the plane. Triangle MNO is determined by the feet of the perpendicular from P to the sides, and triangle QRS is determined by the cevians through P and the circumcircle of triangle ABC.Provethat triangle MNO and QRS are similar.

Crux 462.S6.162. (H.Charles) Soient A, B, C les angles d’un triangle. Montrer que   A tan 2 cos A 1  B  det tan 2 cos B 1 =0. C tan 2 cos C 1 YIU : Problems in Elementary Geometry 292

Crux 463.S6.163*. (J.Garfunkel) Construct an equilateral triangle so that one vertex is at a given point, a second vertex is on a given line, and the third vertex is on a given circle. See also Crux 545.

Crux 464.S6.185.*. (J.C.Fisher and E.L.Koh) (a) If the two squares ABCD and ABCD have vertex A in common and are taken with the same orientation, then the centers of the squares together with the midpoints of BD and BD arethe vertices of a square. (b) What is the analogous theorem for regular n−gons?

Crux 466.S6.188*. (R.Fischler) Soient AB et BC deux arcs d’un cercle tels que arc AB >are BC,etsoitD le point de milieu de l’arc (voir la figure). Si DE ⊥ AB,montrerque AE = EB + BC.[Cethéorème est atribuéàArchiméde].

Crux 472.S6.196. (J.Dou) Construire un triangle connaissant le cote b,lerayonR du cercle circonscrit, et tel que la droite qui joint les centres des cercles inscrit et circonscrit soit parallele au cote a.

Crux 476.S6.217*. (J.Garfunkel) Construct an isosceles right triangle such that the three vertices lie each on one of three concurrent lines, the vertex of the right angle being on the inside line.

Crux 478.S6.219*;11(6)189. (M.S.Klamkin) If the circumcircles of the four faces of a tetrahedron are mutually congruent, then the circumcenter O of the tetrahedron and its incenter I coincide. An editor’s comment following Crux 330 claims that the proof this theorem is easy. Prove it.

Crux 483.S6.226*. (S.Collings) Let ABCD be a convex quadrilateral, AB DC intersect- YIU : Problems in Elementary Geometry 293

ing at F and AD, BC intersecting at G.LetIA, IB, IC , ID bethe incenters of triangles BCD, CDA, DAB,andABC respectively. (a) ABCD is a cyclic quadrilateral if and only if the internal bisectors of the angles at F and G are perpendicular. (b) If ABCD is cyclic, then IAIBIC ID is a rectangle. Is the converse true?

Crux 485.S6.256. (M.S.Klamkin) Given three concurrent cevians of a triangle ABC intesecting at a point P , we construct three new points A, B, C such that AA = k · AP , BB = k · BP, CC = k · CP,wherek>0andk = 1, and the segments are directed. Show that A, B, C, A, B,C lie on a conic if and only if k =2. See also Crux 672.

Crux 488.S6.260*. (K.Satyanarayana) Given a point P within a given angle, construct a line through P such that the segment intercepted by the sides of the angle has minimum length.

Crux 492.S6.291*;7.50,277;8.79. (D.Pedoe) (a) A segment AB and a rusty compass of ≥ 1 span r 2 AB are given. Show how to ﬁnd the vertex C of an equilateral triangleABC,using, as few times as possible, the rusty compass only. 1 (b) Is the construction possible when r<2 AB? See also Crux 592.

Crux 493.S6.294*;7.50. (R.C.Lyness) (a) A, B, C are the angles of a triangle. Prove 1 that there are positive x, y, z,eachlessthan 2 , such that

B C y2 cot +2yz + z2 cot =sinA, 2 2 C A z2 cot +2zx + x2 cot =sinB, 2 2 A B y2 cot +2xy + y2 cot =sinC. 2 2 1 (a) In fact, 2 may be replaced by a smaller k>0.4. What is the least value of k? Note relation to the Malfatti circles.

Crux 504.S7.25*. (L.Bankoﬀ) Given is a triangle ABC and its circumcircle. Find a euclidean construction for a point J inside the triangle such that, when the chords AD, BE, CF are all drawn from J,thenDEF is equilateral.

Crux 506.S7.28*. (M.S.Klamkin) It is known from an earlier problem in this journal YIU : Problems in Elementary Geometry 294

1 1 1 [Crux 14.S1.28] that if a, b, c are the sides of a triangle, then so are b+c , c+a , a+b . Show more generally that if a1, a2, ..., an are the sides of a polygon then for k =1, 2,, ..., n,

n +1 n 1 (n − 1)2 ≥ ≥ , S − a S − a (2n − 3)(S − a ) k i=1,i= k i k where S = a1 + a2 + ···+ an.

Crux 515.S7.57*. (Ngo Tan) Given is a circle with center O and an inscribed triangle ABC.DiametersAA, BB, CC are drawn. The tangent at A meets BC in A, the tangent at B meets CA om B, and the tangent at C meets AB in C. Show that the pointsA, B, C are collinear.

Crux 517.S7.61. (J.Garfunkel)

h h h b + c + a ≤ 3, mc ma mb with equality if and only if the triangle is equilateral.

Crux 520.S7.65*. (M.S.Klamkin) If two chords of a conic are mutually bisecting, prove that the conic cannot be a parabola.

Crux 529.S7.91*. (J.T.Groenman) The sides of a triangle ABC satisfy a ≤ b ≤ c.

◦ sgn(2r +2R − a − b)= sgn(2rc − 2R − a − b)= sgn(C − 90 ). YIU : Problems in Elementary Geometry 295

Crux 535.S7.120*. (J.Garfunkel) Let Ta, Tb, Tc denote the angle bisectors extended to the circumcircle of triangle ABC.Provethat 8√ T T T ≥ 3abc, a b c 9 with equality attained in the equilateral triangle.

Crux 536.S7.122. (B.Leeds) Through each of the midpoints of the sides of a triangle ABC, lines are drawn making an acute angle θ with the sides. These lines intersect to form a triangle ABC.ProvethatABC is similar to ABC and ﬁnd the ratio of similarity.

Crux 540.S7.127*;240. (Leon Bankoﬀ) Professor Euclide Paracelso Bombasto Umbugio has once again retired to his tour d’ivoire where he is now delving into the supersophisticated intricacies of the works of Grassmann, as elucidated by Forder’s Calculus of Extension. His goal is to prove Neuberg’s Theorem: If D, E, F are the centers of squares described externally on the sides of a triangle ABC, then the midpoints of these sides are the centers of squares described internally on thesides of triangle DEF.

Helpthe dedicated professor emerge from his self - imposed conﬁnement and enjoy the thrill of hyperventilation by showing how to solve his problem using only high school, synethetic, euclidean, “plain” geometry.

Crux 544.S7.150*. (N.N.Murty) B C A 2 sin sin ≤ sin 2 2 2 YIU : Problems in Elementary Geometry 296 with equality if and only if the triangle is equilateral. See Klamkin’s solution, relating to AMM S23.

Crux 545.S7.154*. (J.Garfunkel) Given three concentric circles, construct an equilateral triangle having one vertex on each circle.

Crux 548.S7.158. (M.S.Klamkin) If three equal cevians of a triangle divide the sides in the same ratio and same sense, must the triangle be equilateral?

Crux 554.S7.184*;10.197. (G.C.Giri) A sequence of triangle is deﬁned as follows. 0 iis a given triangle, and for each triangle n in the sequence, the vertices of n+1 are the points of contact of the incircle of n with its sides. Prove that n tends to an equilateral triangle as n →∞. See also Crux 463.

Crux 560.S7.243*. (B.C.Rennie) Take a complete quadrilateral. On each of the three diagonals as diameter draw a circle. Prove that these three circles are coaxal.

Crux 562.S7.207. (D.Sokolowsky) Given is a circle γ with center O and diameter of length d,twodistinctpointsP and Q not collinear with O, and a segment of length ,where 0 ≤ ≤ d. Construct a circle through P and Q which meets γ in pointsC and D such that chord CD has length ell.

Crux 565.S7.211*. (J.Garfunkel) In an acute - angled triangle ABC, the altitude issued from vertex A [B, C] meets the internal bisector of angle B [C, A]atP [Q, R]. Prove that

AP · BQ · CR = AI · BI · CI, where I is the incenter of triangle ABC.

Crux 567.S7.214. (G.Tsintsifas) A moving equilateral triangle has its vertices A, B, C on the sides BC, CA, AB respectively of a ﬁxed triangle ABC. The regular tetrahedron MABC has its vertex M always on the same side of the plane ABC. Find the locus of M.

Crux 569.S7.216. (C.W.Trigg) Using euclidean geometry, show that the planes perpendicualr to a space diagonal of a cube at its trisection points contain the vertices of the cube not on that diagonal.

Crux 574.S7.247. (J.Dou) Given ﬁve points A, B, C, D, E, construct a straight line such that the three pairs of straight lines (AD, AE), (BD,BE), (CD,CE) intercept equal segments YIU : Problems in Elementary Geometry 297 on .

Crux 580.S7.253*. (Leon Bankoﬀ) In the ﬁgure, the diameter PQ ⊥ BC and chord AT ⊥ BC. Show that AQ AB + AC TB + TC = = . PQ PB+ PC QB + QC

Crux 584.S7.290;8.16,51*,107;9.23. (F.G.B.Maskell) If a triangle is isosceles, then its centroid, circumcenter, and the center of an escribed circle are collinear. Prove the converse.

Crux 588.S7.306*. (J.Garfunkel) Given is a triangle ABC with internal angle bisectors ta, tb, tc, and medians ma, mb, mc to sides a, b, c respectively. If

ma ∩ tb = P, mb ∩ tc = Q, mc ∩ ta = R, and L, M, N are the midpoints of the sides a, b, c,provethat, AP BQ CR · · =8. PL QM RN See also Crux 685 and 790.

Crux 589.S7.307. (Ngo Tan) In a triangle ABC with seimperimeter s, sides of lengths a, b, c, and medians of lengths ma, mb, mc,provethat (a) There exists a triangle with sides lengths a(s − a), b(s − b), c(s − c). 2 2 2 ma mb mc ≥ 9 (b) a2 + b2 + c2 4 , with equality if and only if the triangle is equilateral. YIU : Problems in Elementary Geometry 298

Crux 592.S7.310. (L.F.Meyers) (a) Given a segment AB of length , and a rusty compass of ﬁxed opening r, show how to ﬁnd a point C such that the length of AC is the mean 1 ≤ ≤ 1 proportiaonal between r and , by use of the rusty compass only, if 4 r , but r = 2 . 1 (b) Show that the construction is impossible if r = 2 . 1 (c) Is the construction possible if r<4 or r>?

Crux 598.S7.316*. (J.Garfunkel) Given a triangle ABC and a segment PQ on side BC, ﬁnd, by euclidean construction, segments RS on side CA and TU on side AB such that, if equilateral triangles PQJ, RSK,andTUL are drawn outside the given triangle, then JKL is an equilateral triangle.

Crux 608.S8.27. (Ngo Tan) ABC is a triangle with sides of lengths a, b, c and semiperimeter s.Provethat 1 1 1 s3 cos4 A +cos4 B +cos4 C ≤ 2 2 2 2abc with equality if and only if the triangle is equilateral.

Crux 609.S8.27*. (J.Garfunkel) A1B1C1D1 is a convex quadrilateral inscribed in a circle and M1, N1, P1, Q1 are the midpoints of the sides B1C1, C1D1, D1A1, A1B1 respectively. The chords A1M1, B1N1, C1P1, D1Q1 meet the circle again in A1, B2, C2, D2 respectively. Quadrilaterals A3B3C3D3 is formed from A2B2C2D2 as the latter was fromed from A1B1C1D1, and the procedure is repeated indeﬁnitely. Prove that quadrilateral AnBnCnDn tends to a square as n →∞.WhathappensifA1B1C1D1 is not convex?

Crux 613.S8.55*,138*. (J.Garfunkel) If A + B + C = 180circ,provethat 1 1 1 2 cos (B − C)+cos (C − A)+cos (A − B) ≥ √ (sin A +sinB +sinC). 2 2 2 3 Here, A, B, C are not necessarily the angles of a triangle, but you may assume that they are if it is helpful to achieve a proof with calculus.

Crux 614.S56. (J.T.Groenman) Given is a triangle with sides of length a, b, c.Apoint P moves inside the triangle in such a way that the sum of the squares of its distances to the three vertices is a constant (= k2). Find the locus of P . 1 2 − 2 2 2 This is the circle, center G,radius 3 3k (a + b + c ).

Crux 615.S8.57. (G.P.Henderson) Let P be a convex n−gon with vertices E1,...,En, perimeter L and area A.Let2θi be the measure of the interior angle at vertex Ei and set YIU : Problems in Elementary Geometry 299

C = cot θi.Provethat L2 − 4AC ≥ 0 and characterize the convex n−gons for which equality holds.

Crux 618.S8.82*,175. (J.A.H.Hunter) For i =1, 2, 3, let Ii be the centers and ri the radii of the three Malfatti circles of a triangle ABC. Calculate the side lengths of the triangle. Comment by Dimitris Vathis, Greece: the inverse Malfatti problem appears and is solved in Pallas, Great Algebra, Athens, 1957 (in Greek), pp.103–104.

Crux 623.S8.91. (J.Garfunkel) If PQRis the equilateral triangle of smallest area inscribed in a given triangle ABC,withP on BC, Q on CA,andR on AB, prove or disprove that AP , BQ, CR are concurrent.

Crux 624.S8.109*. (Dmitry P. Mavlo) ABC is a given triangle of area K,andPQR is the equilateral triangle of smallest area K0 inscribedintriangleABC,withP on BC, Q on CA,andR on AB. K ≡ (a) Find the ratio λ = K0 f(A, B, C) as a function of the angles of the given triangle. (b) Prove that λ attains its minimum value when the given triangle ABC is equilateral. (c) Give a euclidean construction of triangle PQR for an arbitrary given triangle ABC.

Crux 628.S8.115*. (R.H.Eddy) Given a triangle ABC with sides a, b, c,letTa, Tb, Tc denote the angle bisectors extended to the circumcircle of the triangle. If R and r are the cirucm- and in-radii of the triangle, prove that

Ta + Tb + Tc ≤ 5R +2r, with equality just when the triangle is equilateral.

Crux 633.S8.120.*. (J.Aldins, J.S.Kline, and Stan Wagon) It follows from the Wallace - Bolyai - Gerwien Theorem of the early 19th century that any triangle may be cut upinto pieces which may be rearranged using only translations and rotations to form the mirror of the given triangle. This problem once appeared in a Moscow Mathematical Olympiad. Show that such a dissection may be eﬀected with only two straight cuts. YIU : Problems in Elementary Geometry 300

Crux 635.S8.139. (Dan Sokolowsky) In the ﬁgure below, O is the circumcenter of triangle ABC,andPQR ⊥ OA, PST ⊥ OB.Provethat

PQ = QR ⇐⇒ PS = ST.

Crux 637.S8.143*. (J.Bhattacharya) Given a, b, c>0, and 0

a = b cos C + c cos B, b = c cos A + a cos C, c = a cos B + b cos A, prove that a b c = = sin A sin B sin C and A + B + C = π. YIU : Problems in Elementary Geometry 301

crux 643.S8.153. (J.T.Groenman) For i =1, 2, 3, a given triangle has vertices Ai,interior angles αi, and sides ai.SegmentsAiDi, which terminates in ai, bisects angles αi; mi is the perpendicular bisector of AiDi;andEi the intersection of ai and mi.Provethat (a) the three points Ei are collinear; (b) the three segments EiAi are tangent to the circumcircle of the triangle; (c) if pi is the length of AiEi,andifa1 ≤ a2 ≤ a3,then 1 1 1 + = . p3 p1 p2

The points Ei are the centers of the Apollonian circles. The line containing these centers is called the Lemoine axis of the triangle [Court, p.253].

Crux 644.S8.154*. (J.Garfunkel) If I is the incenter of triangle ABC, and the lines AI, BI, CI meet the circumcircle again at D, E, F ,provethat AI BI CI + + ≥ 3. ID IE IF − a b − The ratio AI : ID = b + c a : a.Thesumis b + a 3. See also AMM S23 (1981) 536–537. See also MG 1119, where I is replaced by G, with equality, and MG 1120, where I is replaced by an arbitrary point P , and asks for the set of points for which the inequality holds. YIU : Problems in Elementary Geometry 302

Crux 646.S8.175. (J.C.Fisher) Let M be the midoint of a segment AB. (a) What is the locus of a point the product of whose distances from A and B is the square of its distance from M? (b) In a circle γ through M and B, the three chords BP, BP,andBM satisfy √ BP · BP = 2BM.

Prove that the tangent to γ at M meets the lines BP and BP (extended) in points X and Y , respectively, that are equidistant from M. Note that this fact suggests a construction for the locus of part (a), since X and Y satisfy

XA · XB = YA· YB= XM2.

(c) What is the locus√ of a point for which the absolute value of the diﬀerence of its distances from A and B equals 2 times its distance from M?

Crux 648*.S8.180*. (J.Garfunkel) Given are a triangle ABC, its centroid G,andthe pedal triangle PQR of its incenter I.ThesegmentsAI, BI, CI meet the incircle in U, V , W ; and the segments AG, BG, CG meet the incircle in D, E, F .Let∂ denote the perimeter of a triangle and consider the segment

∂PQR ≤ ∂UV W ≤ ∂DEF.

(a) Prove the ﬁrst inequality. (b) Prove the second inequality.

A A − AI : DI =csc 2 :1;AD : DI =csc2 1 : 1. It follows that A A D =sin [A +(csc − 1)I]. 2 2 YIU : Problems in Elementary Geometry 303

Similarly, B B E =sin [B +(csc − 1)I], 2 2 and C C F =sin [C +(csc − 1)I]. 2 2 The area of triangle DEF is 1 A B C C B A sin sin sin ·ABC[2s + a(csc − 1) + b(csc − 1) + c(csc − 1)] 2s 2 2 2 2 2 2 1 A B C C B A = sin sin sin ·ABC[c csc + b csc + a csc ] 2s 2 2 2 2 2 2 1 B C C A A B = [a sin sin + b sin sin + c sin sin ]. 2s 2 2 2 2 2 2 =

Crux 652.S8.188*. (W.J.Blundon) Let R, r, s represent respectively the circumradius, the inradius, and the semiperimeter of a triangle with angles α, β, γ.Itiswellknownthat s R + r 2rs sin α = , cos α = , tan α = . R R s2 − 4R2 − 4Rr − r2 As for half angles, it is easy to prove that α 4R + r tan = . 2 s α α Find similar expressions for cos 2 and sin 2 . See a paper of Rabinowitz in a later issue (around 1988).

Crux 653.S8.190*. (G.Tsintsifas) For every triangle ABC, show that B − c A cos2 ≥ 24 sin , 2 2 with equality if and only if the triangle is equilateral.

Crux 656.S8.211*. (J.T.Groenman) P is an interior point of a covex region R bounded by the arcs of two intersecting circles C1 and C2. Construct through P a“chord”UV of R,with U on C1 and V on C2, such that PU · PV is a minimum.

Crux 657.S8.212. (Ngo Tan) A quadrilateral ABCD is circumscribed about a circle γ, and AH = AE = a, BE = BF = b, CF = CG = c, DG = DH = d. YIU : Problems in Elementary Geometry 304

Prove that IA a IB b = , = . IC b ID d

Crux 659.S8.215. (Leon Bankoﬀ) If the line joining the incenter I and the circumcenter O of triangle ABC is parallel to the side BC, it is known from Crux 318 that

(2R − r)2(R + r) s2 = . R − r In the same conﬁguration, cos B+cosC = 1, and the internal bisector of angle A is perpendicular to the line joining I to the orthocenter H (MG 758). Prove the following additional properties: (a) If the internal bisector of angle A meets the circumcircle in P , show that AI : IP =cosA. (b) The circumcircle√ of triangle AIH is equal to the incircle of triangle ABC. (c) AI · IP =2Rr = R · AI · BI · CI. 2 B 2 C 1 2 B 2 C 3 (d) sin 2 +sin 2 = 2 ;cos 2 +cos 2 = 2 . 2 A R−r (e) tan 2 = R+r .

Crux 660(corrected 7.274)S8.216. (Leon Bankoﬀ) Show that, in a triangle ABC with semiperimeter s, the line joining the circumcenter and the incenter is parallel to BC if and only if A DL + EM + FN = s tan , 2 where L, M, N bisect the arcs BC, CA, AB of the circumcircle, and D, E, F bisects the sides BC, CA, AB respectively, of the triangle.

Crux 662.S8.218*. (Kaidy Tan) An isosceles triangle has vertex A and base BC. Through apointF on AB, a perpendicular to AB is drawn to meet AC in E and BC produced in D. YIU : Problems in Elementary Geometry 305

With square brackets denoting area, prove synthetically that

[AF E]=2[CDE] ⇐⇒ AF = CD.

Crux 664.S8.220*. (G.Tsintsifas) An isosceles trapezoid ABCD, with parallel bases AB 1 and DC, is inscribed in a circle of diameter AB.ProvethatAC > 2 (AB + DC).

Crux 665.S8.221*,280. (J.Garfunkel) If A, B,C, D are the interior angles of a convex quadrilateral ABCD,provethat √ A + B A 2 cos ≤ cot , 4 2 where the four term sum on each side is cyclic over A, B, C, D, with equality if and only if ABCD is a rectangle.

Crux 666.S8.222. (J.T.Groenman) The symmedians issued from vertices A, B, C of triangle ABC meet the opposite sides in D, E, F respectively. Through D, E, F , lines d, e, f are drawn perpendicular to BC, CA, AB respectively. Prove that d, e, f are concurrent if and only if ABC is isosceles. [For what point P is the cevian triangle also a pedal triangle?]

More generally, if P = f : g : h, the perpendicular from the trace 0 : g; h to the side BC is the line [(a2 + b2 − c2)h − (c2 + a2 − b2)g]x +2abhy − 2acgz =0. Similarly, the equations of the other two perpendiculars can be written down. These three perpendiculars are concurrent if and only if

(b2 + c2 − a2)f(g2 − h2)+(c2 + a2 − b2)g(h2 − f 2)+(a2 + b2 − c2)h(f 2 − g2)=0. YIU : Problems in Elementary Geometry 306

For the symmedian point K, this expression is

(a − b)(b − c)(c − a)(a + b)(b + c)(c + a)(a2 + b2 + c2).

For the incenter, this is (a − b)(b − c)(c − a)(a + b + c)2.

Crux 667.S8.248*. (Dan Sokolowsky) A plane is determined by a line D and a point F C PF not on D.Let denote the conic consisting of all those points P in the plane for which PO = r, where PO is the distance from P to D and r is a given real number. Given a line in the plane, show how to determine by elementary means the intersection (if any) of and C.

Crux 670.S8.251. (O.Bottema) The points Ai, i =1, 2,...,6, no three of which are collinear, are the vertices of a hexagon. X0 is an arbitrary point other than A2 on line A1A2. The line through X0 parallel to A2A3 intersects A3A4 in X1; the line through X1 parallel to A3A6 intersects A6A1 in X2; the line through X2 parallel to A5A6 intersects A4A5 in X3;and the line through X3 parallel to A2A5 intersects A1A2 in X4. (a) Prove the following closure theorem: if X0X1X2X3X4 is closed, i.e., if X4 coincides with X0 for some point X0, then it is closed for any point X0. (b) Show that closure takes place if and only if the six points Ai lie on a conic.

Crux 672.S8.253. (Jordi Dou) Given four points P , A, B, C in a plane, determine points A, B, C on PA, PB, PC respectively, such that AA BB CC = tα, = tβ, = tγ, PA PB PC where α, β, γ are given constants, and such that the hexagon ABCABC is inscribed in a conic. This generalizes Crux 485.

Crux 674(corrected 7.236).S8.256. (G.Tsintsifas) Let ABC be a given triangle and ABC its medial triangle. The incircle of the medial triangle touches its sides in R, S, T .IF the points P and Q divide the perimeter of the original triangle into two equal parts, prove that the midpoint of segment PQ lies on the perimeter of triangle RST .

Crux 675.S8.257. (H.D.Ruderman) ABCD is a skew quadrilateral and P , Q, R, S are points on sides AB, BC, CD, DA respectively. Prove that PR intersects QS if and only if

AP · BQ · CR · DS = PB · QC · RD · SA. YIU : Problems in Elementary Geometry 307

Crux 682.S8.287*;9.23*. (R.C.Lyness) Triangle ABC is acute angled and 1 is its orthic triangle, 2 is the triangular hull of the three excircles aof ABC. Prove that the area of 2 is at least 100 times that of 1.

Crux 683.S8.289*. (Kaidy Tan) Triangle ABC has AB > AC, and the internal bisector of angle A meets BC at T .LetP be any point other than T on line AT , and suppose lines BP, CP intersect lines AC, AB in D, E respectively. Prove that BD > CE or BD < CE according as P lies on the same side or on the opposite side of BC as A.

Crux 685.S8.292.* (J.T.Groenman) Given is a triangle ABC with internal angle bisectors and medians.

Crux 588 asks for a proof of AP BQ CR · · =8. PD QE RF Establish here the inequality AR BP CQ · · ≥ 8, RX PY QZ with equality if and only if the triangle is equilateral. YIU : Problems in Elementary Geometry 308

Crux 686.S8.294*;9.25. (C.W.Trigg) Find the area of the region which is common to four quadrants that have the vertices of a square as centers and a side of the square as a common radius.

√ − π Answer: 1 3+ 3 .

Crux 689.S8.307. (J.Garfunkel) Let ma, mb, mc be the lengths of the medians to sides a, b, c of triangle ABC,andletMa, Mb, Mc denote the lengths of these medians extended to the circumcircle of the triangle. Prove that M M M a + b + c ≥ 4. ma mb mc

Crux 694.S8.314. (J.Garfunkel) Three concurrent circles with radical center R lie inside a given triangle with incenter I and circumcircle O. Each circle touches a pair of sides of the triangle. Prove that O, R,andI are collinear. See also Crux 2137. YIU : Problems in Elementary Geometry 309

Crux 695(corrected 8.30).S8.314. (J.T.Groenman) For i =1, 2, 3, Ai are the vertices of a triangle with sides ai, and excircles with centers Ii touching ai at Bi.Forj, k = i, Mi are the midpoints of BjBk;andmi are the lines through Mi perpendicular to ai. Prove that the mi are concurrent.

The midpoint of B2B3 is the point

M1 =(b + c)(s − a):b(s − b):c(s − c).

The perpendicular to the line BC is

−a(b − c)(a3 − a2b − ab2 + b3 − a2c − 2abc − b2c − ac2 − bc2 + c3)x −b(a3b − a2b2 − ab3 + b4 − a3c − 4a2bc − ab2c +2b3c + a2c2 + abc2 + ac3 − 2bc3 − c4)y +c(−a3b + a2b2 + ab3 − b4 + a3c − 4a2bc + ab2c − 2b3c − a2c2 − abc2 − ac3 +2bc3 + c4)z =0.

The other two perpendiculars can be written down. These three perpendiculars intersect at the point

[a(a5b − a4b2 − 2a3b3 +2a2b4 + ab5 − b6 + a5c +2a2b3c − ab4c − 2b5c − a4c2 YIU : Problems in Elementary Geometry 310

+b4c2 − 2a3c3 +2a2bc3 +4b3c3 +2a2c4 − abc4 + b2c4 + ac5 − 2bc5 − c6)].

Crux 696.S8.316*. (G.Tsintsifas) 3 1 1 − ≥ (a) 4 + 4 cos 2 (B c) cos A; 1 − ≥ − 2r (b) a cos 2 (B C) s(1 R ).

Crux 699.S8.320. (C.W.Trigg) A quadrilateral is inscribed in a circle. One side is a diameter of the circle and the other sides have lengths 3, 4, 5. What is the length of the diameter of the circle?

Crux 700.S9.25.* (Jordi Dou) Construct the center of the ellipse of minimum eccentricity circumscribed to a given convex quadrilateral.

Crux 702.S8.323*;9.144*. (Tsintsifas) Given three distinct points A1,B1,C1 on a circle, and three arbitrary real numbers , m, n adding to 1, show how to determine a point M such that if A1M, B1M, C1M meet the circle again at A, B, C,then

MBC = ABC, MCA = mABC, MAB = nABC. YIU : Problems in Elementary Geometry 311

Crux 703.S9.28. (S.Rabinowitz) A right triangle has legs AB =3andAC =4.Acircle γ with center G is drawn tangent to the two legs and tangent internally to the circumcircle of the triangle, touching that circumcircle in H. Find the radius of γ and prove that GH is parallel to AB.

Crux 708.S9.49 (Murty, Blundon) (a) Prove that (2a − s)(b − c)2 ≥ 0, with equality just when the triangle is equilateral. (b) Prove that the inequality in (a) is equivalent to each of the following 4s a2 ≥ 3( a3 +3abc), s2 ≥ 16Rr − 5r2.

Blundon ﬁrst showed that these inequalities are equivalent, and established a generalization of (1) using Schur’s inequality: if t is any real number and x, y, z > 0, then

xt(x − y)(x − z)+yt(y − z)(y − x)+zt(z − x)(z − y) ≥ 0, with equality if and only if x = y = z.

Crux 709.S9.50. ABC is a triangle with incenter I,andDEF is the pedal triangle of the point I with respect to the sides of ABC. Show that it is always possible to construct four circles each of which is tangent to each of the circumcircles of triangles ABC, EIF, FID, DIE, provided that ABC is not equilateral. YIU : Problems in Elementary Geometry 312

Crux 712.S9.56*. (D.Aitken) Prove that AB = CD in the ﬁgure below.

Crux 715.S9.58. (Murty) Let k be a real number, n an integer. (a) Prove that

8k(sin nA +sinnB +sinnC) ≤ 12k2 +9.

(b) Determine for which k equality is possible in (a), and deduce that √ 3 3 | sin nA +sinnB +sinnC|≤ . 2

Crux 717.S9.64*. (J.T.Groenman and D.J. Smeenk) Let P be any point in the plane of (but not on a side of) triangle ABC.IfHa, Hb, Hc are the orthocenters of triangles PBC, PCA, PAB respectively, prove that [ABC]=[HaHbHc], where the brackets denote the area of triangle.

Editor’s note: This appears in Casey’s Trigonometry, p.146, where the problem is credited to J. Neuberg. YIU : Problems in Elementary Geometry 313

Lemma If ABCDEF is a hexagon (not necessarily convex) such that AB//DE, BC//EF , and CD//F A,then[ACE]=[BDF].

Crux 718.S9.82. (Tsintsifas) ABC is an acute triangle with circumcenter O. The lines AO, BO, CO intersect BC, CA, AB at A1,B1,C1 respectively. Show that 9 OA + OB + OC ≥ . 1 1 1 2

Crux 720.S9.86. (S.Rabinowitz) On the sides AB and AC of a triangle ABC as bases, similar isosceles triangles ABE and ACD are drawn outwardly. If BD = CE, prove or disprove that AB = AC.

Generalization by O.Bottema: If similar triangles ABE and ACD are drawn outwardly, BD = CE if and only if AB = AC or if triangles ABE and ACD are isosceles with vertex at A.

Crux 723.S9.91*,147. (Tsintsifas) Let AG, BG, CG meet the circumcircle again in A,B,C respectively. Prove that ≥ (a) GA + GB + GC AG + BG + CG, AG (b) GA =3, (c) GA · GB · BC ≥ AG · BG · CG.

Crux 724.S9.92*. (H.Ahlburg) Let C + A =2B. Show that (a) sin(A − B)=sinA − sin C. 2 2 (b) a − b = c(a − c). (c) A, C, O, I, H, Ib all lie on a circle, radius R. Furthermore, if this circle meets the lines AB and BC again at A and C,thenAA = CC = |c − a|. (d) OI =√OH (Crux 739 asks for the converse). (e) s = 3(R + r) (Crux 260). YIU : Problems in Elementary Geometry 314

(f) OIb = HIb. (g) The nine-point center N lies on the internal bisector of B.

Crux 727.S9.115*,180*. (J.T.Groenman) Let ta and tb be the symmedians issued from vertices B and C of ABC and terminating in the opposite sides b and c respectively. Prove that tb = tc if and only if b = c. See PME 213.S70S.88.

Crux 728.S9.116*. (S.Rabinowitz) Let E(P, Q, R) denote the ellipse with foci P , Q, which passes through R.IfA, B, C are distinct points in the plane, prove that no two ellipses E(B,C,A), E(C, A, B), and E(A, B, C) can be tangent. See also Crux 1063.

Crux 732.S9.119. (Groenman) Given a ﬁxed triangle and a varying circumscribing triangle determined by angle φ. BC (a) Find a formula for the ratio of similarity λ = λ(φ)= BC . (b) Find the maximal value λm of λ as φ varies in [0,π), and show how to construct the triangle when λ = λm. (c) Prove that λm ≥ 2, with equality nust when the given triangle is equilateral.

Crux 733.S9.121,149,210. (Garfunkel) Let rm be the inradius of the triangle with sides the medians of a given triangle. Prove or disprove 3abc r ≤ , m 4(a2 + b2 + c2) with equality just when the original triangle is equilateral. See also Crux 835.

Crux 735.S9.123*. (S.C.Chan) In a given circle inscribe a triangle so that two sides may pass through two given points and the third side be parallel to a given straight line.

Crux 739.S9.153*,210. (G.C.Giri) Prove that if I is equidistant from O and H, then one oftheanglesis60◦.

Crux 743.S9.182. (G.Tsintsifas) A point M lies on the disc ω with diameter OG,. The lines AM, BM, CM meet the circumcircle again in A,B,C respectively, and G is the centroid of ABC.Provethat (a) M does not lie in the interior of disc ω with diameter OG; (b) ABC ≤ABC. YIU : Problems in Elementary Geometry 315

Crux 745.S9.187*. (R.Izard) In the adjoining ﬁgure, ABC and DEF are both equilateral, and angles BAD, CBE, ACF are all equal. Prove that the triangles ABC and DEF have the same center. See also MG 1161.

Crux 746.S9.187*. (J.Garfunkel) Given are two concentric circles and a triangle ABC inscribed in the outer circle. A tangent to the outer circle at A is rotated about A in the counterclockwise sense until it ﬁrst touches the inner circle, say at P . The procedure is repeated at B and C, resulting in points A and R respectively, on the inner circle. Prove that PQR is directly similar to ABC.

Crux 747.S9.188. (J.T.Groenman) Let ABC be a triangle inscribed in a circle with center O.ThesegmentsBC, CA, AB are divided internally in the same ratio by points A1, B1, C1 so that BA1 : A1C = CB1 : B1A = AC1 : C1B = λ : µ, where λ + µ = 1. A line through A1 perpendicular to OA meets the circle in two points, one of which, Pa, lies on the arc CAB, and other points Pb and Pc are determined analogously by lines through B1 and C1 perpendicular to OBand OC.Provethat

2 2 2 2 2 2 APa + BPb + CPc = a + b + c , independently of λ and µ. Investigate the situation if the word “internally” is replaced by “externally”. YIU : Problems in Elementary Geometry 316

Crux 749.S9.190,280* ( R.R.Tiwari) Solve the system of equations

yz(x + y + z)(y + z − x) = a2, (y + z)2 zx(x + y + z)(z + x − y) = b2, (z + x)2 xy(x + y + z)(x + y − z) = c2. (x + y)2

Crux 752.S9.212. (Groenman) For i =1, 2, 3, let Ai be the vertices of a triangle with angles αi,sidesai, circumcenter O and inscribed circle γ. The lines AiO intersect γ in Pi and Qi. (a) Prove that

P1Q1 : P2Q2 : P3Q3 = f(cos α1):f(cos α2):f(cos α3), where f(x) is a function to be determined. (b) Prove or disprove that α2 = α3 if and only if P2Q2 = P3Q3.

Crux 755.S9.244*. (L.Csirmaz) Find the locus of points with coordinates

(cos A +cosB +cosC, sin A +sinB +sinC)

(a) if A, B, C are real numbers with A + B + C = π, (b) if A, B, C are angles of a triangle.

Crux 756.S9.217. (Lu YANG and Jingzhong ZHANG) Given three vertices of a parallelogram, ﬁnd the fourth vertex using only a rusty compass.

Crux 757.S9.218. (A.Aeppli) Given only two distinct points A and B, prove or disprove that the midpoint of the segment AB can be found using only a rusty compass.

Crux 758.S9.218. (S.Rabinowitz) Find necessary and suﬃcient coondition for the roots of a cubic equation to be the vertices of an equilateral triangle.

Crux 759.S9.221. (J.Garfunkel) Given are four congruent circle intersecting in a point O, and a quadrilateral ABCD circumscribing these circles with each side of the quadrilateral tangent to two circles. Prove that the quadrilateral ABCD is cyclic. YIU : Problems in Elementary Geometry 317

Crux 760.S9.247. (Jordi Dou) Given ABC, construct on AB and AC directly similar isosceles trinagles ABX and ACY such that BY = CX. Prove that there are exactly two such pairs of isosceles triangles.

Crux 762(corrected 8.278);S9.249. (J.T.Groenman) The internal bisectors of a triangle meet the opposite sides in D, E, F respectively. The area of DEF is . (a) Prove that 3abc 1 ≤ ≤ . 4(a3 + b3 + c3) 4 3abc 5 (b) If a =5and 4(a3+b3+c3) = 24 , determine b and c, given that they are integers.

Crux 766.S9.254*. (S.Rabinowitz) Let ABC be an equilateral trinagle with center O. Prove that if P is a variable point on a ﬁxed caircle with center O, then the triangle whose sides have length PA,PB,PC has a constant area.

Crux 768.S9.282. (J.Garfunkel)

4 B − C 2 sin B sin C ≤ cos ≤ cos A. 9 2 3

Crux 770.S9.285. (K.Satyanarayana) Let P be an interior point of triangle ABC.Prove that PA· BC + PB · CA > PC · AB.

Crux 777.S10.20*. (Bottema) Let Q = ABCD be a convex quadrilateral with sides AB = a, BC = b, CD = c, DA = d,andarea[Q].√ The following theorem is well known: If Q has both a circumcircle and an incircle, then [Q]= abcd. √ Prove or disprove the following converse: If Q has a circumcircle and [Q]= abcd,then there exists a circle tangent to the four lines AB, BC, CD,andDA.

Crux 778.S9.318. (J.T.Groenman) Let ABC be a triangle with incenter I, the lines AI, BI, CI meeting its circumcircle again in D, E, F respectively. If S is the sum and P the product of the numbers ID IE IF , , , AI BI CI prove that 4P − S =1. YIU : Problems in Elementary Geometry 318

Crux 782.S10.25*. (H.S.M.Coxeter) (a) Sketch the plane cubic curve given by the para- metric equations

x = α(β − γ)2,y= β(γ − α)2,z= γ(α − β)2,α+ β + γ =0, where (x, y, z) are barycentric coordinates, referred to an equilateral triangle. In what respect do its asymptotes behave diﬀerently from those of a hyperbola? (b) Eliminate the parameters α, β, γ to obtain a single equation

x3 + y3 + z3 + a(x2y + xy2 + y2z + yz2 + z2x + x2z)+bxyz =0 for certain numbers a and b. (c) What equation does the curve have in polar coordinates?

Crux 786.S10.55*. (O.Bottema) Let r − 1, r2, r3 be arbitrarily chosen positive numbers. Prove that there exists a real triangle whose exradii are r1, r2, r3, and calculate the sides of this triangle.

Crux 787.S10.56*. (J.W.Lynch) (a) Given two sides, a and b, of a triangle, what should be the length of the third side, x, in order that the area enclosed be a maximum? (b) Given three sides, a, b, c, of a quadrilateral, what should be the length of the fourth side, x, in order that the area enclosed be a maximum? See also Crux 914, on the interpretation of the positive root of

x3 − (a2 + b2 + c2)x − 2abc =0.

Crux 790.S10.60. (R.H.Eddy) Let ABC be a triangle with sides a, b, c in the usual order, and let a, b, c and a, b, c be two sets of concurrent cevians, with a, b, c intersecting a, b, c in L, M, N respectively. If

∩ ∩ ∩ a b = P, b c = Q, c a = R,

prove that, independently of the choice of concurrent cevians a, b, c,wehave AP BQ CR abc · · = ≥ 8, PL QM RN BL · CM · AN with equality occurring just when a, b, c are the medians of the triangle. This problem extends Crux 588. YIU : Problems in Elementary Geometry 319

Crux 795.S10.92. (J.Garfunkel) Given a triangle ABC,letta, tb, tc be the lengths of its internal angle bisectors, and let Ta, Tb, Tc be the lengths of these bisectors extended to the circumcircle of the triangle. Prove that 4 T + T + T ≥ (t + t + t ). a b c 3 a b c Solution. (B.Prielipp) 4s 4√ √ √ √ 4 Ta + Tb + Tc ≥ √ ≥ s( s − a + s − b + s − c) ≥ (t1 + tb + tc). 3 3 3 The ﬁrst inequality was established by Groenman in his solution to Crux 628.S8.114, and the last two follow from item 8.9 of the Bottema Bible, where they are credited to Santal´o (1943).

Crux 805.S10.120. (Klamkin) If x, y, z > 0, prove that x + y + z yz + zx + xy ≥ √ , 3 y2 + yz + z2 + z2 + zx + x2 + x2 + xy + y2 with equality if and only if x = y = z.

Crux 806.S10.121*. (K.Satyanarayana) Let LMN be the cevian triangle of the point S for triangle ABC. It is trivially true that S is the centroid of ABC =⇒ S is the centroid of LMN. Prove the converse.

Crux 808**.S10.132;11(1)18. (S.Rabinowitz) Find the length o the largest circular arc ontained within the right triangle with sides a ≤ b

Crux 812.S10.128. (D.Sokolowsky) Let C be a given circle, and let Cim i =1, 2, 3, 4, be circles such that (i) Ci is tangent to C at Ai,fori =1, 2, 3, 4; (ii) Ci is tangent to Ci+1 for i =1, 2, 3. Furthermore, let be a line tangnet to C at the other extremity of the diameter of C through A1,andfori =2, 3, 4, let A1Ai intersect at Pi. Prove that, if C, C1,andC4 are ﬁxed, then the ratio of unsigned lengths P2P3 : P3P4 is constant for all circles C2 and C3 that satisfy (i) and (ii).

Crux 814.S10.131*. (Leon Bankoﬀ) Let D denote the point BC but by the internal hisector of angle BAC in the Heronian triangle whose sides are c =14,a =13,b = 15. With D as center,√ describe the circle touching AC in L and cutting the extension of AD in J. Show AJ 5+1 that AL = 2 , the golden ratio. YIU : Problems in Elementary Geometry 320

Crux 815.S10.132*. (J.T.Groeman) Prove the triangle inequalities √ 1 4s 3 ≥ , awa abc √ 1 2s awa ≥ 3 3 4 3 , awa (abc) with equality if and only if the triangle is equilateral.

Crux 816 (Tsintsifas) Prove the triangle inequalities ≤ (b + c) 8R(R +2r)s, ≤ bc(b + c) 8R(R + r)s, a3 ≤ 8s(R2 − r2).

Crux 818.S10.159*. (A.P.Guinand) Let ABC be a scalene triangle with P the point where the internal bisector of A intersects the Euler line. If O, H, P are given, construct angle A, using only ruler and compass.

Crux 825.S10.168. (J.Garfunkel) Of the two triangle inequalities

A A tan2 ≥ 1, and 2 − 8 sin ≥ 1, 2 2 the ﬁrst is well known and the second is equivalent to the the well known inequality A 1 sin ≤ . 2 8 Prove or disprove the sharper inequality A A tan2 ≥ 12 − 8 sin . 2 2

Crux 827.S10.199*. (J.T.Groenman) For i =1, 2, 3, let Ai be the vertices of a triangle with opposite sides ai,letBi be an arbitrary point on ai,andletMi be the midpoint of BjBk. If the lines bi are perpendicular to ai through Bi, and if the lines mi are perpendicular to ai through Mi, prove that the bi are concurrent if and only if the mi are concurrent. YIU : Problems in Elementary Geometry 321

What are the coordinates of the intersection of the lines mi,giventhatB1B2B3 is the pedal triangle of a point P ?

Crux 829.S10.201. (Bottema) Prove that the signed area of GIO is given by

(a + b + c)(b − c)(c − a)(a − b) − . 48

Crux 835.S10.227*. (J.Garfunkel) Let Rm be the circumradius of the trinagle formed by medians. Prove that a2 + b2 + c2 R ≥ . m 2(a + b + c) See also Crux 733 and 970.

Crux 836.S10.228*,320*. (V.N.Murty) (a)

(1 − cos A)(1 − cos B)(1 − cos C) ≥ cos A cos B cos C, with equality if and only if the triangle is equilateral. (b) Deduce Bottema’s triangle inequality

(1 + cos 2A)(1 + cos 2B)(1 + cos 2C)+cos2A cos 2B cos 2C ≥ 0.

Crux 844.S10.264. (P.M.Gibson) (a) A triangle A0B0C0 with centroid G0 is inscribed in a circle Γ with center O. The lines A0G0, B0G0, C0G0 meet Γ again in A1, B1, C1, respectively, and G1 is the centroid of triangle A1B1C1. A triangle A2B2C2 with centroid G2 is obtained in the same way from A1B1C1, and the procedure is repeated indeﬁnitely, producing triangles with centroids G3, G4, .... If gn = OGn, prove that the sequence {g0,g1,g2,...} is descreasing and converges to zero. (b) Prove or disprove that a result similar to (a) holds for a tetrahedron inscribed in a sphere, or more generally, for an n−simplex inscribed in an n−sphere.

Crux 845.S10.266*. (B.M.Saler) Let r1,r2,r3 be the focal radii (all from the same focus 2 2 √ x y 3 F) of the points P1,P2,P3 on the ellipse a2 + b2 = 1. A circle center F and radius r = r1r2r3 intersects the focal radii in P1, P2, P3 respectively. Find the ratio of the areas of P1P2P3 and P1P2P3. (This is Theorema Elegantissimum from Acta Eruditorum, A.D. 1771, p.131, author unknown.) YIU : Problems in Elementary Geometry 322

Crux 846.S10.267. (J.Garfunkel) Prove the triangle inequalities

mambmc ≥ 2 2 2 r, ma + mb + mc ≥ 2 12Rmambmc a(b + c)ma, 4R ama ≥ bc(b + c), 1 m 2R ≥ a . bc mbmc

r s Crux 850.S10.271. (V.N.Murty) Let x = R and y = R .Provethat √ √ y ≥ x( 6+ 2 − x), with equality if and only if the triangle is equilateral.

Crux 852.S10.296. (Jordi Dou) Given are three distinct points A, B, C on a circle. A point P in the plane has the property that if the lines PA, PB, PC meet the circle again in A, B, C respectively, then AB = AC. Find the locus of P .

R−2r Crux 856.S10.303*. (Garfunkel) For a triangle ABC,letM = 2R . An inequality P ≥ Q involving elements of the triangle will be called strong or weak respectively, according as

P − Q ≤ M or P − Q ≥ M.

(a) Prove that the inequality α 3 sin2 ≥ 2 4 is strong. (b) Prove that the inequality α cos2 ≥ sin β sin γ 2 is weak.

Crux 858.S10.306. (J.T.Groenman) For n =0, 1, 2,...,letPn be a point in the plane whose distances from the sides satisfy 1 1 1 d : d : d = : : . a b c an bn cn

(a) A point Pn being given, show how to construct Pn+2. YIU : Problems in Elementary Geometry 323

(b) Using (a), or otherwise, show how to construct the point Pn for an arbitrary given value of n.

Crux 859.S10.307;11(6)191. (V.N.Murty) Let ABC be a triangle of type II, namely, ≤ ≤ π ≤ r ≥ 1 α β 3 γ. It is known that for such a trinagle x := R 4 . Prove the stronger inequality √ 3 − 1 x ≥ . 2

Crux 862.S10.322. (G.Tsintsifas) P is an interior point of ABC,LinesthroughP parallel to the sides of the triangle meet those sides in the points A1, A2, B1, B2, C1, C2 as shown in the ﬁgure. Prove that ≤ 1 (a) A1B1C1 3 ABC, ≥ 2 (b) A1C2B1A2C1B2 3 ABC.

Crux 865.S10.325*. (C.Kimberling) Let x, y, z be the distances from the sides to a variable point P inside a triangle. Prove that if 0 = t = 1, the critical point of xt + yt + zt satiﬁes x : y : z = ap : bp : cp 1 where p = t−1 . Discuss limiting cases.

Crux 866.S10.327*. (Jordi Dou) Given a triangle ABC with sdies a, b, c, ﬁnd the minimum value of a · XA + b · XB + c · XC,whereX ranges over all the points of the plane of the triangle.

Crux 872(corrected 10.18)S10.334. (G.Tsintsifas) Let P be a point other than a vertex in the plane of a triangle ABC. It is known that there exists a triangle with sides a · PA, b · PB, c · PC.IfR0 is the circumradius of this triangle, prove that

PA· PB· PC ≤ RR0. When does equality hold ?

Crux 876.S10.339.√ √ (J.T.Groenman)√ Let Ka, Kb, Kc be the circles with centers A, B, C and radii λ bc, λ ca, λ ab. Find the locus of the radical center of Ka, Kb, Kc as λ ≥ 0.

Crux 880**.S11(1)26*. (C.Kimberling) For a given triangle ABC, what curve is formed by all the points P in three-dimensional space satisfying

BPC = CPA = AP B? YIU : Problems in Elementary Geometry 324

Crux 883.S11(1)28*. (J.Tabov and S.Troyanski) Let ABC be a triangle with area S, sides a, b, c,mediansma, mb, mc, and interior angle bisectors ta, tb, tc.If

ta ∩ mb = P, tb ∩ mc = Q, tc ∩ ma = R, σ 1 prove that S < 6 ,whereσ is the area of triangle PQR.

c · A + b · B + c · C a · A + c · B + a · C b · A + a · B + b · C P = ,Q= ,R= . b +2c c +2a a +2b Note that the coordinates of R on p.29 is misprinted. The ratio [PQR] bc2 + ca2 + ab2 − 3abc = . [ABC] (b +2c)(c +2a)(a +2b) See also Curx 588, 585, and 790.

Crux 890.S11(2)55. (L.F.Meyers) Construct triangle ABC, with straightedge and compass, given the lengths of b an c of two sides, the midpoint Ma of the third side, and the foot Ha of the altitude to that third side.

Crux 892.S11(2)57*. (Stan Wagon) ABCD is a square and ECD an isosceles triangle with base angles 15◦, as shown in the ﬁgure. Prove that AEB =60◦ (and therefore triangle AEB is equilateral). This problem is very well known, but all published solutions use trigonometry and/or auxiliary lines. What is required is a simple proof with trigonometry or any auxiliary lines (or circles).

Proof. (K.S.Williams) Write AEB =2x,and BEC = y, in degrees. Then, BAE =90− x, and 2x ≥ 60 ⇐⇒ x ≥ 30 ⇐⇒ BC ≤ BE ⇐⇒ AB ≤ BE ⇐⇒ 2x ≤ 90 − x ⇐⇒ 2x ≥ 60. YIU : Problems in Elementary Geometry 325

Therefore, AEB =2x = 60. Editor’s remark: Such a proof can actually be found in Coxeter and Greitzer, Geometry Revisited, pp.25, 158.

Crux 895.S11(2)60. (J.T.Groenman) Let ABC be a triangle with sides a, b, c in the usual order and circumcircle Γ. A line through C meets the segment AB in D, Γ again in E, and the perpendicular bisector of AB in F . Assume that c =3b. (a) Construct the line for which the length of DE is maximal. (b) If DE has maximal length, prove that DF = FE. (c) If DE has maximal length and also CD = DF, ﬁnd a in terms of b and the measure of angle A.

Crux 896.S11(2)62. (J.Garfunkel)

A 1 B − C 3 sin2 ≥ 1 − cos ≥ . 2 4 2 4

Crux 902.S11(3)86. (J.C.Fisher) (a) For any point P on a side of a given triangle, deﬁne Q to be that point on the triangle for which PQ bisects the area. What is the locus of the midpoint of PQ? (b) Like the curve in part (a), the locus of the midpoints of the perimeter - bisecting chords of a triangle (see Crux 674) has an orientation that is opposite to that of the given triangle. Is this a general principle? More precisely, given a triangle and a family of chords joining P (t)to Q(t), where (i) P (t)andQ(t) move counterclockwise about the triangle as t increases and (ii) P (t) = Q(t) for any t, does the midpoint of PQ always trace a curve that is clockwise oriented?

Crux 903.S11(3)88. (S.Rabinowitz) Let ABC be an acute - angled triangle with circumcenter O and orthocenter H. (a) Prove that an ellipse with foci O and H can be inscribed in the triangle. (b) Show how to construct, with straightedge and compass, the points L, M, N where this ellipse is tangent to the sides BC, CA, AB of the triangle. (c) Prove that AL, BM, CN are concurrent.

Crux 904.S11(3)90. (G.Tsintsifas) Let M be any point in the plane of triangle ABC. The cevians AM, BM, CM intersect the lines BC, CA, AB in A, B, C respectively. Find the locus of the point M such that

[MCB]+[MAC]+[MBA]=[MCB]+[MAC]+[MBA]. YIU : Problems in Elementary Geometry 326

Crux 905.S11(3)91. (J.T.Groenman) Let ABC be a triangle that is not right angled at B or C.LetD be the foot of the perpendicular from A upon BC,andletM and N be the feet of the perpendiculars from D upon AB and AC, respectively. (a) Prove that, if A =90circ,then BMC = BNC. (b) Prove or disprove the converse of (a).

Crux 908.S11(3)93*. (M.S.Klamkin) Determine the maximum value of

P =sinα A sinβ B sinγ C, where A, B, C are the angles of a triangle, and α, β, γ are given positive numbers.

Crux 910.S11(3)96*. (O.Bottema) Determine the locus of the centers of the conics through the incenter and the three excenters of a given triangle. Pedoe pointed out that this appears in Durell’s Projective Geometry, p.203, and wrote

The incenter is the orthocenter of the triangle formed by the three excenters, so that all conics through the four points are rectangular hyperbolas, and the locus of the centers of these conics is the nine-point circle of the triangle formed by any three of the four points, and this is the circumcircle of the original triangle.

Editor’s remark: the locus of the foci of all parabolas tangent to the three sides of a triangle is the circumcircle of the triangle. (W.P. Milne, Homogeneous Coordinates, (1924) p.118).

Crux 919.S11(4)131. (J.Dou) Show how to construct a point P which is the centroid of triangle ABC where A, B, C are the orthogonal projections of P upon the three given lines a, b, c respectively. Solution. The symmedian point is the only point which is the centroid of its own pedal triangle.

Problem Find the point P which is the circumcenter of its cevian triangle.

Crux 920.S11(4)131. (B.C.Rennie) If a triangle of unit base and unit altitude is in the unit square. Show that the base of the triangle must be one side of the triangle.

Crux 923.S11(5)150. (C.Kimberling) Let the ordered triple (a, b, c) denote the triangle whose side lengths are a, b, c. Similarity being an equivalence relation on the set of all triangles, let the ordered ratios a : b : c (which we call a triclass) denote the equivalence class of all triangles (a,b,c) such that a : a = b : b = c : c. YIU : Problems in Elementary Geometry 327

Let T be the set of all triclass. A multiplication ◦ on T is deﬁned by

a : b : c ◦ α : β : γ = aα +(c − b)(γ − β):bβ +(a − c)(α − γ):cγ +(b − a)(β − α).

(a) Prove that (T,◦) is a group. (b) If Tˆ is the set of all a : b : c such that a, b, c are integers, prove that every triclass in Tˆ is a unique product of “prime” triclasses.

Crux 925.S11(5)154*. (J.T.Groenman) The points Ai, i =1, 2, 3, are the vertices of a triangle with sides ai and median lines mi. Through a point P , the lines parallel to mi intersects ai in Si. Find the locus of P if the three points Si, i =1, 2, 3 are collinear. See also Crux 1073. Solution. Let P = xA + yB + zC in barycentric coordinates. The point X is P + k(−2A + − x B+C)forsomek.Thisis(x 2k)A+(y+k)B+(z+k)C,andk = 2 if this lies on BC.Thepoint x x y y X is therefore (y + 2 )B +(z + 2 )C. Similarly, Y and Z are the points Y =(x + 2 )A +(z + 2 )C z z and Z =(x + 2 )A +(y + 2 )B. These are collinear if and only if   0 x +2yx+2z det  2x + y 0 y +2z  =0. 2x + z 2y + z 0

This reduces to (x + y + z)(xy + yz + zx) = 0. The locus is therefore the Steiner ellipse. AX, BY , CZ are collinear if and only if (x − y)(y − z)(z − x)=0.

Crux 926.S11(5)155. (S.Rabinowitz) Let P be a ﬁxed point inside an ellipse, L avariable chord through P ,andL the chord through P that is perpendicular to L.IfP divides L into two segments of lengths m and n,andifP divides L into two segments of lengths r and s, 1 1 prove that mn + rs is a constant.

Crux 929.S11(5)159. (K.Satyanarayana) Given a triangle ABC, ﬁnd all interior points P such that, if AP , BP, CP meet the circumcircle again in A1, B1, C1, then triangles ABC and A1B1C1 are congruent. YIU : Problems in Elementary Geometry 328

Crux 934.S11(6)194. (Leon Bankoﬀ) As shown in the ﬁgure, the diameter AB,avariable chord AJ, and the intercepted minor arc JB of a circle (O) form a mixtilinear triangle whose inscribed circle (W ) touches arc JB in K and whose mixtilinear excircle (V ) touches arc JB in L. The projections of W and V upon AB are C and D respectively. As J moves along the circumference of circle (O), the ratio of the arcs KL and LB varies. (a) When arcs KL and LB are equal, what are their values? (b) Show that BD is equal to the side of the inscribed square lying in the right angle of triangle ADV .

Crux 937.S11(6)199. (Jordi Dou) ABCD is a trapezoid in a cirlceφ,withAB//DC.The midpoint of AB is M, and the line DM meets the circle again in P . A line through P meets the lines BC in A, CA in B, AB in C, and the circle again in F .Provethat(AB,CF )is a harmonic range.

Crux 939.S11(7)224. (G.Tsintsifas) ABC is an acute triangle with AB < AC,and orthocenter H. M being an interior point of segment DH, lines BM and CM intersect sides CA and AB in B and C respectively. Prove that BB

Crux 940.S11(7)226. (J.Garfunkel)

7 A 9 sin B sin C ≤ +4 sin ≤ . 4 2 4 YIU : Problems in Elementary Geometry 329

Crux 947.S11(7)233. (Jordi Dou) Let ABCD be a quadrilateral (not necessarily convex) with AB = BC, CD = DA,andAB ⊥ BC. The midpoint of CD being M,pointsK and L are found on line BC such that AK = AL = AM.IfP , Q, R are the midpoints of BD, MK, ML respectively, prove that PQ ⊥ PR.

Crux 1038.S12.216. (J.Dou) Given are two concentric circles and two lines through their centers. Construct a tangent to the inner circle such that one of its points of intersection with the outer circle is the midpoint of the segment of the tangent cut oﬀ by the two given lines.

Crux 1039.S13.152;14.176. (K.Satyanarayana) Given are three collinear points O, P , H (in that order) such that OH < 3OP. Construct a triangle ABC with circumcenter O and orthocenter H and such that AP is the internal bisector of angle A. How many such triangles are possible?

Crux 1059.S12(10)290. (Kimberling) In his book, The Modern Geometry of the Triangle, (London, 1913), W.Gallatly denotes by J the circumcenter of triangle I1I2I3, whose vertices are the excenters of the reference triangle ABC. On pages 1 and 21 are ﬁgures in which J appears to be collinear with the incenter and the circumcenter of triangle ABC. Are these points really collinear?

Crux 1062.S13(1)17. (Klamkin) (a) Let Q be a convex quadrilateral inscribed in a circle with center O.Prove: (i) If the distance of any side of Q from O is half the length of the opposite side, then the diagonals of Q are orthogonal. (ii) Conversely, if the diagonals of Q are orthogonal, then the distance of any side of Q from O is half the length of the opposite side. (b)∗ Suppose a convex quadrilateral Q inscribed in a centrosymmetric region with center O satisﬁes either (i) or (ii). Prove or disprove that the region must be a circle.

Crux 1064.S13(1)22. (G.Tsintsifas) Triangles ABC and DEF are similar, with angles EF A = D, B = E, C = F , and ratio of similarity λ = BC . Triangle DEF is inscribed in triangle ABC,withD, E, F on the lines BC, CA, AB, not necessarily respectively. Three cases can be considered. Case 1: D, E, F on BC, CA, AB respectively; Case 2: D, E, F on CA, AB, BC respectively; Case 3: D, E, F on AB, BC, CA respectively. ≥ 1 For case 1, it is known that λ 2 [See Crux 606]. Prove that, for each of cases 2 and 3, λ ≥ sin ω,whereω is the Brocard angle of triangle ABC. YIU : Problems in Elementary Geometry 330

Crux 1073.S13(2)56. (J.Dou) Let K be an interior point of triangle ABC. Through a point P in the plane of the triangle, parallels to the cevians AK, BK, CK are drawn to meet BC, CA, AB at L, M, N respectively. If the points L, M, N are collinear, (a) prove that the locus of P is an ellipse; (b) construct the center of this ellipse; See also Crux 925, which is the special case when K = G.

Crux 1074.S13(2)59. (J.T.Groenman) Let ABC be a triangle with circumcenter O. Prove that (a) there are two points P in the plane of the triangle such that

PA2 : PB2 : PC2 =sec2 A :sec2 B :sec2 C;

(b) these two points and O are collinear; (c) these two points are inverse with respect to the circumcircle of the triangle.

Crux 1075.S13(2)60*. (J.Dou) Let P be an interior point of triangle ABC.Denotebyρ and ρ the inradii of triangle ABC and the pedal triangle of P .Provethat 1 OP ≥ OI ⇒ ρ ≤ ρ. 2 Give an example to show the converse does not hold. [Notation changed and statement corrected].

Crux 1076.S13(2)62. (M.S.Klamkin) Let x, y, z denote the distances from an interior point of a given triangle ABC to the respective vertices A, B, C;andletK be the area of the pedal triangle of P with respect to ABC. Show that

x2 sin 2A + y2 sin 2B + z2 sin 2C +8K is a constant independent of P .

Crux 1090.S13(4)126*. (Dan Sokolowsky) Let Γ be a circle with center O,andA a ﬁxed point distinct from O in the plane of Γ. If P is a variable point on Γ and AP meets Γ again in Q, ﬁnd the locus of the circumcenter of triangle POQ as P ranges over Γ.

Crux 1091.S13(4)128. (Kimberling) Let A1A2A3 be a triangle and γi the excircle opposite Ai, i =1, 2, 3. Apollonius knew how to construct the circle Γ internally tangnet to the three excircles and encompassing them. Let Bi be the point of contact of Γ and γi, i =1, 2, 3. prove that the lines A1B1, A2B2, A3B3 are concurrent. YIU : Problems in Elementary Geometry 331

Crux 1100.S13(5)160;(8)258. (D.J. Smeenk) ABC is a triangle with C =30◦,circumcenter O and incenter I.PointsD and E are chosen on BC and AC respectively, such that BD = AE = AB.ProvethatDE = OI and DE ⊥ OI. See also Crux 1196.

Crux 1103.S13(5)163. (R. Izard) Three concurrent cevians through the vertices A, B, C of a triangle meet the lines BC, CA, AB in D, E, F respectively, and the internal bisector of angle A meets BC in V .IfA, F , D, V , E are all concyclic, prove that AD ⊥ BC.

Crux 1106.S13(5)167. (J.Garfunkel) The directly similar triangles ABC and DEC are both right angled at C.Provethat (a) AD ⊥ BE; AD (b) BE equals the ratio of similitude of the two triangles. Pedoe: The problem can be generalized (the triangles need not be right-angled), but (b) is incorrectly stated. The generalized statement is: The directly similar triangles ABC and DEC have angle Ω at C.Provethat (a) AD makes angle Ω with BE; AD CA CD (b) BE = CB = CE .

Crux 1109.S13(10)322*;14(3)78. (D.J. Smeenk) ABC is a triangle with orthocenter H. A rectangular hyperbola with center H intersects line BC in A1 and A2, line CA in B1 and B2, and line AB in C1 and C2. Prove that the midpoints of A1A2, B1B2 and C1C2 are collinear.

Crux 1140.S13(6)232. (J.Dou) Given triangle ABC, construct a circle which cuts (extended) lines BC, CA, AB in pairs of points A and A, B and B, C and C respectively such that angles AAA, BBB and CCC are all right angles.

Crux 1170.S13.332. (C.Kimberling) In the plane of triangle ABC,letP and Q be points having trilinears α1 : β1 : γ1 and α2 : β2 : γ2, respectively, where at least one of the products α1α2, β1β2, γ1γ2 is nonzero. Give a euclidean construction for the point P ∗ Q having trilinears α1α2 : β1β2 : γ1γ2. (A point has trilinears α : β : γ if its signed distances to the sides BC, CA, AB are respectively proportional to the numbers α, β, γ.)

Crux 1171.S13.333. (D.S.Mitrinovic and J.E.Pecaric) (i) Determine all real numbers λ so that, whenever a, b, c are the lengths of three segments which can form a triangle, the same is true for (b + c)λ,(c + a)λ,(a + b)λ.[Forλ = −1, we have Crux 14.S1.28]. (ii) Determine all pairs of real numbers λ, µ so that, whenever a, b, c are the lengths of three segments which can form a triangle, the same is true for (b+c+µa)λ,(c+a+µb)λ,(a+b+µc)λ. YIU : Problems in Elementary Geometry 332

Crux 1174.S14(1)17. (Kimberling) Suppose ABC is an acute triangle. Prove that there is an point inside ABC and points D, E on BC, F , G on CA,andH, I on AB such that GP H, IPD,andEPF are congruent equilateral triangles.

Crux 1177.S14(1)20. (Tsintsifas) ABC is a triangle and M an interior point with barycentric coordinates (λ1,λ2,λ3). Lines HMD, JMF, EMI are parallel to AB, BC, CA respectively. The centroids of triangles DME, FMH, IMJ are denoted by G1, G2, G3 respectively. Prove that 1 [G G G ]= (λ λ + λ λ + λ λ )[ABC]. 1 2 3 3 1 2 2 3 3 1

Crux 1180.S14(1)24**. (J.R.Pounder) (a) It is well known tht the Simson line of a point P on the circumcircle of a triangle T envelopes a deltoid (Steiner’s hypocycloid) as P varies. Show that this is true of an oblique Simson line as well. (An oblique Simson line of T is the line passing through the points A1, B1, C1 chosen on edges BC, CA, AB respectively so that the lines PA1, PB1, PC1 make equal angles (say θ) in the same sense of rotation, with BC, CA, AB respectively. The usual Simson line occurs when θ =90circ. (b*) Given such an oblique deltoid for T , locate a triangle T similar to T such that the “normal” deltoid for T and the oblique deltoid for T coincide.

Crux 1184.S14(1)29. (S.Rabinowitz) Let ABC be a nonequilateral triangle and let O, I, H, F denote the circumcenter, incenter, orthocenter, and the centero fo the nine-point circle, respectively. Can either of the triangles OIF or IFH be equilateral?

Crux 1188.S14(1)32. (D.Sokolowsky) Given a circle K and distinct points A, B in the plane of K, construct a chord PQ of K such that B lies on the line PQ and PAQ =90◦.

Crux 1191.S14(2)55. (H.Fukagawa) Let ABC be a triangle, and let points D, E, F be on sides BC, CA, AB respectively such that triangles AEF , BFD,andCDE all have the same inradius r.Letr1 and r2 denote the inradii of DEF and ABC respectively. Show that r + r1 = r2.

Crux 1192.S14(2)56. (R.K.Guy) Let ABC be an equilateral triangle and v, w be arbitrary positive real numbers. S (respectively T , U) is the Apollonius circle which is the locus of points whose distances from A and B (respectively A and C, B and C) are in the ratio v : w (respectively v : v + w, w : v + w). Prove that S, T , U have just one point in common, and that it lies on the circumcircle of triangle ABC.

Crux 1195.S14(2)62. (Kimberling) Let ABC be a triangle with medians ma, mb, mc and circumcircle Γ. Let DEF be the triangle formed by the parallels to BC, CA, AB through A, YIU : Problems in Elementary Geometry 333

B, C respectively, and let Γ be the circumcircle of DEF.LetA, B, C be the triangle formed by the tangents to Γ at the points (other than A, B, C)wherema, mb, mc meet Γ. Finally let A , B , C be the points (other than D, E, F )wherema, mb, mc meet Γ . prove that the lines AA, BB, CC concur in a position on the Euler line of triangle ABC.

Crux 1196.S14(2)62. (J.Dou) Let I be the incenter and O the circumcenter of triangle ABC.LetD on AC and E on BC be such that AD = BE = AB.ProvethatDE is perpendicular to OI. See also Crux 1100.

Crux 1198.S14(3)85,(6)179,(10)318. (Groenman) Let ABC be a triangle with incenter I, Gergonne point G, and Nagel point N.LetJ be the isotomic conjugate of I.ProvethatG, N, J are collinear.

Crux 1203.S14.91. (M.N.Naydenov) A quadrilateral inscribed in a circle of radius R and circumscribed around a circle of radius r has consecutive sides a, b, c, d, semiperimeter s and area F .Provethat√ √ (a) 2 F ≤ s ≤ r = r2 +4R2; √ (b) 6F ≤ ab + ac + ad + bc + bd + cd ≤ 4r2√+4R2 +4r r2 +4R2; (c) 2sr2 ≤ abc + abd + acd + bcd ≤ 2r[r + r2 +4R2]2; 2 ≤ ≤ 16 2 2 2 (d) 4Fr abcd 9 r (r +4R ).

Crux 1216.S14.120;21.131. (W.Janous) Prove or disprove that √ sin A sin B sin C 9 3 2 < + + ≤ . A B C 2π

Crux 1217.S14.123;20.297. (N.Bejlegaard) Given are two lines 1 and 2 intersecting at A,andapointP inthesameplane,whereP does not lie on either angle bisector at A.Also given is a positive real number r. (a) Construct a line through P , intersecting 1 and 2 at B and C respectively, such that AB + AC = r. (b) Construct a line through P , intersecting 1 and 2 at B and C respectively, such that |AB − AC| = r.

Crux 1224.S14(5)145,(8)236. (Tsintsifas) A1A2A3 is a triangle with circumcircle Ω. Let x1

(b) X1 + X2 + X3 ≥ 3(x1 + x2 + x3) ≥ 12r,wherer is the inradius of triangle A1A2A3.

Crux 1239.S14(6)181*. (J.T.Groenman) Find all points whose pedal triangles with respect to a given triangle are isosceles and right-angled.

Crux 1243.S14(6)187. (Tsintsifas) Let ABC be a triangle and M an interior point with barycentric coordinates (λ1,λ2,λ3). The distances of M from the vertices A, B, C are x1, x2, x3, and the circumradii of triangles MBC, MCA, MAB, ABC are R1, R2, R3, R. Show that

λ1R1 + λ2R2 + λ3R3 ≥ R ≤ λ1x1 + λ2x2 + λ3x3.

Crux 1252.S14(7)211*. (Tsintsifas) Let ABC be a triangle and M an interior point with barycentric coordinates λ1,λ2,λ3. We denote the pedal triangle and the cevian triangle of M by DEF and ABC respectively. Prove that

[DEF] s ≥ 4λ λ λ ( )2, [ABC] 1 2 3 R where s is the semiperimeter and R the circumradius of triangle ABC.

Crux 1260.S14(8)236*;15(2)51. (H.Fukagawa) Let ABC be a triangle with angles B and C acute, and let H be the foot of the perpendicular from A to BC.LetO1 be the circle intrnally tangent to the circumcircle O of triangle ABC and touching the segments AH and BH.LetO3 be the circle similarly tangnet to O, AH,andCH. Finally, let O2 be the incircle of triangle ABC, and denote the radii of O1, O2, O3 by r1, r2, r3 respectively. Show that r1+r3 (a) r2 = 2 ; (b) the centero of O1, O2, O3 are collinear.

Crux 1269.S14(9)270. (Janous) Let ABC be a non-obtuse triangle with circumcenter M and circumradius R.Letu1, u2, u3 be the lengths of the parts of the cevians (through M) between M and the sides opposite to A, B, C respectively. Prove or disprove that R u + u + u ≤ 1 2 3

Crux 1272.S14(8)256. (J.T. Groenman) Let A1A2A3 be a triangle. Let the incircle have center I and radius ρ, and meet the sides of the triangle at points P1, P2, P3.LetI1, I2, I3 be the excenters and ρ1, ρ2, ρ3 the exradii. Prove that (a) the lines I1P1, I2P2, I3P3 concur at a point S; YIU : Problems in Elementary Geometry 335

(b) the distances d1, d2, d3 of S to the sides of the triangle satisfy

d1 : d2 : d3 = ρ1 : ρ2 : ρ3.

a b c This is the point X57 = s−a : s−b : s−c .

Crux 1273.S14(9)276. (Tsintsifas) Let ABC be a triangle, M an interior point, and ABC its pedal triangle. Denote the sides of the two triangles by a, b, c,anda, b, c respectively. Prove that a b c + + < 2. a b c

Crux 1275.S14(9)279. (P.Penning) On a circle C with radius R three points A1, A2, A3 are chosen arbitrarily. Prove that the three circles with radius R, not coinciding with C,and passing through two of the points A1, A2, A3 intersect in the orthocenter of triangle A1A2A3.

Crux 1279.S14(9)284*. (Dou) Consider a triangle whose orthocenter lies on its incircle. (a) Show that if one of its angles is given, the others are determined. (b) Show that if it is isosceles, then its sides are in the proportion 4:3:3.

Crux 1280.S14(9)287. (Janous) Let ABC be a triangle and let A1, B1, C1 be points on BC, CA, AB respectively, such that A C B A C B 1 = 1 = 1 k>1. BA1 CB1 AC1 Show that k2 − k +1 perimeter (A B C ) k < 1 1 1 < , k(k +1) perimeter (ABC) k +1 and that both bounds are best possible.

Crux 1282.S14(10)305. (Tsintsifas) Let ABC be a triangle, I the incenter, and A, B, C the intersections of AI, BI, CI with the circumcircle. Show that

IA + IB + IC − (IA + IB + IC) ≤ 2(R − 2r).

Crux 1290.S14(10)314. (J.B.Tabov) The triangles B1B2B3 and C1C2C3 are homothetic and each of them is in perspective with the triangle A1A2A3 (vertices with the same index correspond). Di (i =1, 2, 3) is the midpoint of the segment BiCi. Prove that triangle A1A2A3 and D1D2D3 are in perspective. YIU : Problems in Elementary Geometry 336

Comment by Jordi Dou: The proposition is false!

Crux 1295.S15(1)17*. (J.T.Groenman) Let A1A2A3 be a triangle with I1, I2, I3 the excenters and B1, B2, B3 the feet of the altitudes. Show that the lines I1B1, I2B2, I3B3 concur at a point collinear with the incenter and circumcenter of the triangle. 3 2 2 2 2 This is the point X46 =[a(a + a (b + c) − a(b + c ) − (b + c)(b − c) )]. Kimberling notes that this is a known result (Casey, Analytic Geometry, 2nd ed., Hodges & Figgis, Dublin, 1893, p.85).

Crux 1305.S15(1)31*. (J.T.Groenman) Let A1A2A3 be an acute triangle with circumcenter O.LetP1, Q1 denote the intersection of A1O with A2A3 and with the circumcircle respectively. Deﬁne P2, Q2, P3, Q3 analogously. Prove that OP1 · OP2 · OP3 ≥ (a) P1Q1 P2Q2 P3Q3 1; OP1 OP2 OP3 ≥ (b) P1Q1 + P2Q2 + P3Q3 3; A1P1 · A2P2 · A3P3 ≥ (c) P1Q1 P2Q2 P3Q3 27.

Crux 1307.S15(2)58. (Dou) Let A, B, C be the intersections of the bisectors of triangle ABC with the opposite sides, and let A, B, C be the midpoints of BC, CA, AB respectively. Prove that AA, BB, CC are concurrent.

Crux 1309.S15(2)61. (Kimberling) Let ABC be a triangle with circumcircle Γ, and let DEF be the triangle formed by the lines tangent to Γ at A, B, C. Call a triangle ABC a circumcevian triangle if for some point P , A is the point other than A where AP meets Γ, and similarly for B and C.ProvethatDEF is perspective with every circumcevian triangle.

Crux 1315.S15(3)88. (J.T.Groenman) Let ABC be a triangle with medians AD, BE, CF and median point G. We denote triangles AGF , BGF, BGD, CGD, CGE, AGE by i, i =1, 2, 3, 4, 5, 6 respectively. Let Ri and ri denote the circumradius and inradius of i.Prove that (i) R1R3R5 = R2R4R6; 15 1 1 1 1 1 1 9 (ii) 2r < r1 + r3 + r5 = r2 + r4 + r6 < r .

Crux 1317.S15(3)91;16(3)80. (A.Bondesen) Crux 1133 suggests the following problem. In a triangle ABC the excircle touching side AB touches line BC and AC at points D and E respectively. If AD = BE, must the triangle be isosceles?

Crux 1321.S15(4)116*;16(3)81. (Dou) The circumcircle of a triangle is orthogonal to an excircle. Find the ratio of their radii. Answer: R : rc =1:2. YIU : Problems in Elementary Geometry 337

Crux 1329.S15(4)126*. (D.J. Smeenk) Let ABC be a triangle, and let congruent circles C1, C2, C3 be tangent to half lines AB and AC, BA and BC, CA and CB, respectively. (a) Determine the locus of the circumcenter P of triangle DEF,whereD, E, F are the centers C1, C2, C3. (b) If C1, C2, C3 all pass through the same point, show that 1 1 1 = + , ρ r R where r and R are the inradius and the circumradius of triangle ABC.

Crux 1342.S15(6)188*. (Groenman) Let ABC be a triangle and let D and E be the midpoints of BC and AC respectively. Suppose that DE is tangent to the incircle of triangle ABC.Provethatrc = r,wherer is the inradius of triangle ABC and rc the exradius to AB.

Crux 1343.S15(6)189*. (D.J. Smeenk) ABC is an acute triangle and D, E are the feet of the altitudes to BC, AC respectively. Suppose DE is tangent to the incircle. Show that rc =2R,whereR is the circumradius and rc is the exradius to AB.

Crux 1355.S15(8)240;16(3)81. (Tsintsifas) Let ABC be a triangle and I its incenter. The perpendicular to AI at I intersects the line BC at the point A. Analogously, we deﬁne B and C.ProvethatA, B, C lie in a straight line.

Crux 1359.S15(8)244. (G.R. Veldkamp) Let PQR, PST,andPUV be congruent isosceles triangles with common apex P andhavingnovertexincommonotherthanP . The sense P → Q → R, P → S → T ,andP → U → V is anticlockwise. We suppose moreover that VQ and RS meet in A, RS and TU meet in B,andTU and VQin C.ProvethatP is on the line joining the circumcenter to the symmedian point of triangle ABC.

Crux 1372.S15(9)284*. (D.J. Smeenk) Triangle ABC has circumcenter O and median point G, and the line AG and BG intersect the circumcircle again at A1 and B1 respectively. Suppose that A, B, O, G are concyclic. Show that (a) AA1 = BB1; (b) triangle ABC is acute angled.

Crux 1376.S15(9)287. (Veldkamp) Let ABCD be a quadrilateral with an inscribed circle of radius r and a circumscribed circle of radius R.LetAC = p and BD = q be the diagonals. Prove that pq 4R2 − =1. 4r2 pq YIU : Problems in Elementary Geometry 338

Crux 1379.S15(10)307. (P.Penning) Given are an arbitrary triangle ABC and an arbitrary interior point P . The pedal-points of P on BC, CA, AB are D, E, F respectively. Show that hte normal from A to EF,fromB to FD,andfromC to DE are concurrent.

Crux 1385.S16(1)20. (Klamkin) Show that the sides of the pedal triangle of any interior point of an equilateral triangle T are proportional to the distances from P to the corresponding vertices of T .

Crux 1391.S16(1)28. (Tsintsifas) Let ABC be a triangle and D the point on BC so that theincircleoftriangleABD and the excircle (to side DC) of triangle ADC have the same radius ρ1. Deﬁne ρ2 and ρ3 analogously. Prove that 9 ρ + ρ + ρ ≥ r, 1 2 3 4 where ρ is the inradius of triangle ABC.

Crux 1393.S16(2)45. (J.T.Groenman) Let A1A2A3 be a triangle with incenter I, excenters IA, IB, IC , and median point G.LetH1 be the orthocenter of triangle I1A2A3, and deﬁne H2 and H3 analogously. Prove that A1H1, A2H2, A3H3 are concurrent at a point collinear with G and I.

Crux 1395.S16(2)46,(10)299. (Janous) Given an equilateral triangle ABC, ﬁnd all points P inthesameplanesuchthatPA2, PB2, PC2 form a triangle. YIU : Problems in Elementary Geometry 339

Crux 1404.S16(3)83. (J.T.Groenman and D.J. Smeenk) Let ABC be a triangle with circumradius R and inradius ρ. A theorem of Poncelet states that there is an inﬁnity of triangles having the same circumcircle and the same incircle as triangle ABC. (a) Show that the orthocenters of these triangles lie on a circle. (b) If R =4ρ, what can be said about the locus of the centers of the nine-point circles of these triangles?

Crux 1421.S16(4)123. (J.T.Groenman) ABC is a triangle with sides a, b, c. The excircle to the side a has center Ia and touches the sides at D, E, F . M is the midpoint of BC. (a) Show that the lines IaD, EF,andAM are concurrent at a point Sa. (b) In the same way we have points Sb and Sc.Provethat 3 [S S S ] > [ABC]. a b c 2

Crux 1425.S16(5)146;17(6)175. (Dou) Let D be the midpoint of side BC of the equilateral triangle ABC and ω a circle through D tangnet to AB, cutting AC in points B1 and B2. YIU : Problems in Elementary Geometry 340

Prove that the two circles, distinct from ω, which pass through D and are tangnet to AB,and which respectively pass through B1 and B2, have a point in common on AC.

Crux 1430.S16(5)158*;17(2)48*. (M.Bencze) AD, BE, CF are (not necessarily concurrent) cevians in triangle ABC, intersecting the circumcircle of triangle ABC in the pointsP , Q, R.Provethat AD BE CF + + ≥ 9. DP EQ FR When does equality hold? Same as Math. Mag. 1402 (June 1993).

Crux 1432.S16(6)180*;17(1)18. (J.T.Groenman) If the Nagel point of a triangle lies on the incircle, prove that the sum of two of the sides of the triangle equals three times the third side.

Crux 1436.S16(6)186*. (D.J. Smeenk) ApointP lies on the circumcircle Γ of a triangle ABC, P not coinciding with one of the vertices. Circles Γ1 and Γ2 pass through P and are tangent to AB at B,andtoAC at C respectively. Γ1 and Γ2 intersect at P and at Q. (a) Show that Q lies on the line BC. (b) Show that as P varies over Γ the line PQ passes through a ﬁxed point on Γ.

Crux 1437.S16(6)187. (Tsintsifas) Let ABC be an equilateral triangle inscribed in a triangle ABC,sothatA ∈ BC, B ∈ CA, C ∈ AB.If

BA CB AC = = , AC BA CB prove that triangle ABC is equilateral.

Crux 1442.S16(6)191. (J.T.Groenman) Let ABC be a triangle. If P is a point on the circumcircle, and D, E, F are the feet of the perpendiculars from P to BC, AC, AB respectively, then it is well known that D, E, F are collinear. Find P such that E is the midpoint of the segment DF.

Crux 1444.S16(7)214. (Dou) Given the center O of a conic γ and three points A, B, C lying on γ, construct those points X on γ such that XB is the bisector (interior or exterior) of angle AXC.

Crux 1446.S16(7)217. (Tsintsifas) Let ABC be an equilateral triangle inscribed in triangle ABC,sothatA ∈ BC,etc.DenotebyG, G the centroid, by O, O the circumcenters, YIU : Problems in Elementary Geometry 341 by I, I the incenters, and by H, H the orthocenters of triangles ABC and ABC respectively. Prove that in each of the four cases (a) G = G, (b) O = O, (c) I = I, [not correct; Seimiya gave a counterexample]. (d) H = H, ABC must be equilateral.

Crux 1453.S16(8)247. (D.J. Smeenk) Triangle ABC movesinsuchawaythatAB passes through a fixed point P and AC passes through a fixed point Q. Prove that throughout the motion, BC is tangent to a fixed circle.

Crux 1455.S17(8)249. (Tsintsifas) Let ABC be a triangle inscribed in triangle ABC, so that A ∈ BC, etc. Suppose that BA CB AC = = =1 , AC BA CB and that triangle ABC is similar to triangle ABC. Prove that the triangles are equilateral.

Crux 1464.S16(9)282. (Tsintsifas) Let ABC be a triangle inscribed in triangle ABC, so that A ∈ BC,etc. (a) Prove that BA CB AC = = AC BA CB if and only if the centroids G, G of the two triangles coincide. (b) Prove that if (1) holds, and either the circumcenters O, O or the orthocenters H, H of the triangles coincide, then triangle ABC is equilateral. (C)* If (1) holds and the incenters I and I of the triangle coincide, characterize triangle ABC.

Crux 1466.S16(9)285. (J.T.Groenman and D.J. Smeenk) On the sides of triangle A1A2A3 and outside the triangle we draw similar triangles A3A2B1, A1A3B2 and A2A1B3 with geocenters G1, G2,andG3 respectively. The geocenters of triangles A1B3B2, A2B1B3, A3B2B1 and A1A2A3 are Γ1,Γ2,Γ3,andG respectively. It is known that G is the geocenter of triangle B1B2B3 as well. [See Math. Mag. 50 (1985) 84 – 89]. Show that Γ1G1 has midpoint G,length 2 | | 3 A1B1 , and is parallel to A1B1.

Crux 1471.S16(10)304. (Tsintsifas) Let ABC be an equilateral triangle inscribed in a triangle ABC,sothatA ∈ BC, B ∈ CA, C ∈ AB,andsothatABC and ABC are directly similar. If BA = CB = AC, prove that the triangle are equilateral. [Not correct!] YIU : Problems in Elementary Geometry 342

Crux 1476.S16(10)309. (K.R.S.Sastry) A triangle is called self - altitude if it is similar to the triangle formed from its altitudes. Suppose triangle ABC is self - altitude, with sides a ≥ b ≥ c and angles bisectors AP , BQ, CR. Prove that the lengths of CP, PB, BR. RA form a geometric progression.

Crux 1480.S16(10)316. (J.B.Romero M´arquez) ABC and ABC are triangles connected by a dilatation (BC//BC, CA//CA, AB//AB)andA = BC ∩ BC, B = AC ∩ AC, C = AB ∩ AB). Show that triangle ABC is connected to either of the two given triangles by a dilatation, and that the centroids of the three triangles are collinear.

Crux 1483.S17(1)22*. (Tsintsifas) Let ABC be an equilateral triangle inscribed in a triangle ABC,sothatA ∈ BC, B ∈ CA, C ∈ AB,andsothatABC and ABC are directly similar. (a) Show that, if the centroids G and G of the triangles coincide, then either the triangles are equilateral or A, B,C are the midpoints of the sides of triangle ABC. (b) Show that if either the circumcenters O, O or the incenters I, I of the triangles coincide, then the triangles are equilateral.

Crux 1486.S17(2)50. (Dou) Given three triangles T1, T2, T3 and three points P1, P2, P3, construct points X1, X2, X3 such that the triangles X2X3P1, X3X1P2 and X1X2P3 are directly similar to T1, T2, T3 respectively.

Crux 1491.S17(1)30. (J.T.Groenman) In triangle ABC, the internal bisector of angle A meet BC at D, and the external bisector of angles B and C meet AC and AB (produced) at E and F respectively. Suppose that the normals to BC, CA, AB at D, E, F , respectively, meet. Prove that AB = AC.

Crux 1492.S17(2)50*;25(8)508*. (Tsintsifas) Let ABC be an equilateral triangle inscribed in a triangle ABC,sothatA ∈ BC, B ∈ CA, C ∈ AB. Suppose also that BA = CB = AC. (a) If either the centroids G, G or the circumcenters O, O of the triangles coincide, prove that triangle ABC is equilateral. (b)* If either the incenters I, I or the orthocenters H, H of the triangles coincide, characterize triangle ABC.

Crux 1510.S17.91;19.50,204;21.159. (J.Garfunkel) Let P be any point inside triangle ABC.LinePA, PB, PC are drawn and angles PAC, PBA, PCB are denoted by α, β, γ YIU : Problems in Elementary Geometry 343 respectively. Prove or disprove that A B C cot α +cotβ +cotγ ≥ cot +cot +cot , 2 2 2 with equality when P is the incenter of triangle ABC.

Crux 1637.S18.125;20.165;24(7)427. (G.Tsintsifas) Prove that sin B +sinC 12 > A π for a non-obtuse triangle.

Crux 1722.S19(2)56. (Seimiya) ABCD is a cyclic quadrilateral with BD < AC.LetE and F be the intersections of AB, CD and of BC, AD, respectively, and let L and M be the midpoints of AC and BD.Provethat LM 1 AC BD = − . EF 2 BD AC

Crux 1730.S19.81;20.18. (G.Tsintsifas) Prove that in ABC, 1 bc(s − a)2 ≥ sabc. 2

Crux 1740.S19.94,305;21.159*. (Dan Pedoe) In triangle ABC the points N, L, M,in that order on AC, are respectively the foot of the perpendicular from B to AC, the intersection with AC of the bisector of angle ABC, and the midpoint of AC. The nagles ABN, NBL, LBM and MBC are all equal. Determine the angles of triangle ABC.

Crux 1756.S19.6.172. (K.R.S.Sastry) For positive integers n ≥ 3andr ≥ 1, the n−gonal number of rank r is deﬁned as r2 r P (n, r)=(n − 2) − (n − 4) . 2 2 Call a triple (a, b, c) of natural numbers, with a ≤ b ≤ c,ann−gonal Pythagorean triple if P (n, a)+P (n, b)=P (n, c). When n = 4, we get the usual Pythagorean triple. (i) Find an n−gonal Pythagorean triple for each n. (ii) Consider all triangles ABC whose sides are n−gonal Pythagorean triples for some n ≥ 3. Find the maximum and minimum possible values of angle C. YIU : Problems in Elementary Geometry 344

Remark: See also S. Hirose, Fibonacci Quarterly, 24 (1986) 99 – 106.

Crux 1812.S20.20. (T.Seimiya) ABC is a right triangle with right angle at C.LetD be apoint on side AB,andletM be the midpoint of CD. Suppose that AMD = BMD.Prove that ACD : BCD = CDA : CDB.

Crux 1814.S20.23*. (D.J. Smeenk) Given are the fixed line l with two fixed points A and B on it, and a fixed angle ϕ. Determine the locus of the point C with the following property: the angle between l and the Euler line of ABC is equal to ϕ.

Crux 1820.S20.29. (J.B.Romero Márgque) Let O be the point of intersection of the diagonals AC and BD of the quadrangle ABCD. Prove that the orthocenters of the four triangles OAB, OBC, OCD, ODA are the vertices of a parallelogram that is similar to the figure formed by the centroids of these four triangles. What if ‘centroids’ is replaced by ‘circumcenter’ ? bc ≥ Crux 1827.S20.57;21.54*;22.36*,78. (S.Arslanagoć and M.M.Milssevi’c) (i) A(s−a) 12s π . (ii) It follows easily from the proof of Crux 1611 and the correction on 19.79 that also

b + c 12s ≥ . A π Do the two summations above compare in general? a ≥ 9 Crux 1843.S20.113;21.55*. (S.Arslanago´c and M.M.Milssevi’c) (i) 2A(s−a) π . (ii) It is obvious that 1 9 ≥ . A π Do these two summations compare in general?

Crux 1895.S20.263;21.204*. (J.Chen and G.Yu) Let P be an interior point of triangle A1A2A3; R1, R2, R3 the distances from P to A1, A2, A3;andR the circumradius of triangle A1A2A3.Provethat 32 R R R ≤ R3, 1 2 3 27 with equality when A2 = A3 and PA2 =2PA1. YIU : Problems in Elementary Geometry 345

Crux 1902.S (Kuczma) ABC is a triangle with circumcircle Γ. Let P be a variable point on the arc ACB of Γ, other than A, B, C. X and Y are points on the rays AP abd BP respectively such that AX = AC and BY = BC. Prove that the line XY always passes through aﬁxedpoint. See also Crux 1993.

Crux 1904.S21.204*. (K.W.Lau) Prove that

2 2 2 ma(bc − a )+mb(ca − b )+mc(ab − c ) ≥ 0.

Crux 1906.S (K.R.S.Sastry) Let AP bisect angle A of triangle ABC,withP on BC.Let Q be the point on segment BC such that BQ = CP.Provethat

AQ2 = AP 2 +(b − c)2.

Crux 1908.S20(10)293. (C.J.Bradley) In ABC the feet of the perpendiculars from A, B, C onto BC, CA, AB are denoted by D, E, F respectively. H is the orthocenter. The triangle is such that all of AH − HD, BH − HE and CH − HF are positive. K is an interal point of ABC and L, M, N are the feet of the perpendiculars from K onto BC, CA, AB respectively. Prove tht AL, BM, CN are concurrent if KL : KM : KN is equal to (i) AH − HD : BH − HE : CH − HF; 1 1 1 (ii) AH−HD : BH−HE : CH−HF . YIU : Problems in Elementary Geometry 346

1995 – 1999

Crux 1910.S21.22 (J.Kotani) The octahedron ABCDEF is inscribed in a sphere so that the three diagonals AF , BD, CE meet at a point, and the centroids of the six triangular faces of the octahedron are also inscribed in a sphere. Show that (i) the orthocneters of the six faces are inscribed in a sphere; (ii) (AB · DF = AD · BF)(AC · EF + AE · CF)(BC · DE + CD · BE)=36V 2, where V is the volume of the octahedron.

Crux 1912.S21.26. (T.Seimiya) ABC is a triangle with AB = AC. Similar triangles ABD and ACE are drawn outwardly on the sides AB and AC of ABC,sothat ABD = ACE and BAD = CAE. CD and BE meet AB and AC at P and Q respectively. Prove that AP = AQ if and only if [ABD] · [ACE]=[ABC]2, where [XY Z] denotes the area of triangle XY Z. (This problem is an extension of Crux 1537.)

Crux 1914.S21.28*. (K.R.S.Sastry) Let A1A2 ···An be a regular n−gon, with M1, M2, ..., Mn the midpoints of the sides. Let P be a point in the plane of the n−Gon. Prove that 180◦ PM ≥ cos PA . i n i

Crux 1918.S21.34*. (D.J. Smeenk) ABC is a triangle with circumcenter O and incenter I,andK, L, N are the midpoints of BC, CA, AB respectively. Let E and F be the feet of the altitudes from B and C respectively. (a) If OK2 = OL2 + OM2, show that E, F , O are collinear and determine all possible values of BAC. (b) If instead OK = OL + OM, show that E, F , I are collinear, and determine all possible values of BAC.

Crux 1920.S21.58*. (W.Janous) Let a, b, c be the sides of a triangle. (a) Prove that for any 0 <λ≤ 2, 1 (a + b)(b + c)(c + a) 2 3 < ≤ , (1 + λ)2 (λa + b + c)(a + λb + c)(a + b + λc) 2+λ and that both bounds are best possible. (b) What are the bounds for λ>2? YIU : Problems in Elementary Geometry 347

Crux 1921.S21.61. (T.Seimiya) D and E are points on sides AB and AC of triangle ABC such that DE//BC,andP is an interior point of ADE. PB and PC meet DE at F and G respectively. Let O1 and O2 be the circumcenters of PDG and PFE respectively. Prove that AP ⊥ O1O2.

Crux 1923.S21.64*. (K.R.S.Sastry) In triangle ABC, cevians AD, BE, CF are equal and concur at point P .Provethat

PA+ PB + PC =2(PD+ PE + PF).

Crux 1926.S21.67. (W.Pompe) On sides BC, CA, AB of triangle ABC are chosen points A1, B1, C1 respectively, such that triangle A1B1C1 is equilateral. Let o1, o2, o3 and O1, O2, O3 be respectively the incircles and the incenters of triangle AC1B1, BA1C1, CB1A1.If O1C1 = O2C1, show that (a) B1O3 = B1O1 and A1O2 = A1O3; (b) three external common tangents to the pairs of circles o1,o2, o2,o3, o3,o1 diﬀerent from the sides of triangle ABC, have a common point.

Crux 1930.S21.72. (V.Konecny) T1 is an isosceles triangle with circumcircle K.LetT2 be another isosceles triangle inscribed in K whose base is one of the equal sides of T1,andwhihc overlaps the interior of T1. Similarly create isosceles triangles T3 from T2, T4 from T3,andso on. Do the triangles Tn approach an equilateral triangle as n →∞?

Crux 1931.S21.92. (T.Seimiya) M is the midpoint of side BC of ABC,and Γ is the circle with diameter AM. D and E are the other intersections of Γ with AB andAC respectively. Let P be the point such that PD and PE are tangent to Γ. Prove that PB = PC. YIU : Problems in Elementary Geometry 348

Crux 1933.S21.96*. (G.Tsintsifas) Two externally tangent circles of radii R1 and R2 are internally tangent to a semicircle of radius 1, as in the ﬁgure. Prove that √ R1 + R2 ≤ 2( 2 − 1).

Crux 1935.S21.108*. (M.S.Klamkin) Given an ellipse which is not a circle, prove or disprove that the locus of the midpoints of suﬃciently small constant length chords is another ellipse.

Crux 1937.S21.102*. (D.J. Smeenk) Triangle ABC has circumcenter O, orthocenter H, and altitudes AD, BE,andCF (with D on BC etc). Suppose OH//AC. (a) Show that EF, FD,andDE are in arithmetic progression. (b) Determine the possible values of angle B.

Crux 1939.S21.105. (C.J.Bradley) Let ABC be an acute-angled triangle with circumcenter O, incenter I, and orthocenter H.LetAI, BI, CI meet BC, CA, AB at U, V , W ,and AH, BH, CH meet BC, CA, AB respectively in D, E, F .ProvethatO is an interior point of triangle UVW if and only if I is an interior point of triangle DEF.

Crux 1941.S21.133. (T.Seimiya) ABCD is a convex quadrilateral, and O is the intersection of its diagonals. Suppose that the area of the (nonconvex) pentagon ABOCD is equal to the area of triangle OBC.LetP and Q be the points on BC such that OP//AB and OQ//DC. Prove that [OAB]+[OCD]=2[OPQ], where [XY Z] denotes the area of triangle XY Z.

Crux 1943.S21.134*. (K.R.S.Sastry) In triangle ABC, the median AD is the geometric YIU : Problems in Elementary Geometry 349 mean of AB and AC.Provethat √ 1+cosA = 2| cos B − cos C|.

Crux 1947.S21.139. (D.J. Smeenk) Triangle ABC has incenter I and centeroid G.The line IG intersects BC, CA, AB in K, L, M respectively. The line through K parallel to CA intersects the internal bisector of angle BAC at P . The line through L parallel to AB intersects the internal bisector of angle CBA in Q. The line through M parallel to BC intersects the internal bisector of angle ACB in R. Show that BP, CQ, AR are parallel.

Crux 1949.S21.141. (F.Ardila) Let D, E, F be points on the sides BC, CA, AB respectively of triangle ABC,andletR be the circumradius of ABC.Provethat 1 1 1 AB + BC + CA ( + + )(DE + EF + FD) ≥ . AD BE CF R

Crux 1951.S21.163. (T.Seimiya) ABCD is a cyclic quadrilateral, and P is the intersection of the diagonals AC and BD. A line through P meets AB and CD at E and F respectively. Let O1 and O2 be the circumcenters of PAB and PCD,andletQ be the point on O1O2 such that PQ ⊥ .ProvethatEP : PF = OQ : QO2.

Crux 1952.S21.164*. (K.R.S.Sasstry) The convex cyclic quadrilateral ABCD is such that each of its diagonals bisects one angle and trisects the opposite angle. Determine the angles of ABCD.

π ≤ Crux 1954.S21.166*. (V.N.Murty) Let ABC be a triangle with A< 2 and B C. The tangents to the circumcircle of triangle ABC at B and C meet at D.Putθ = OAD, where O is the circumcenter. Prove that

2tanθ =cotB − cot C. YIU : Problems in Elementary Geometry 350

Crux 1956.S21.169*. (G.Tsintsifas) In a semicircle of radius 4, there are√ three tangent circles as in the ﬁgure. Prove that the radius of the smallest circle is at most 2 − 1.

Crux 1960.S21.174*. (W.Pompe) Two perpendicular linesand a circle pass through a common point. Three line segments AB, CD, EF, with endpoints on the two perpendicular lines, are drawn tangent to C at their midpoints. Prove that the length of one segment is equal to the sum of the lengths of the other two.

Crux 1961.S21.175. (T.Seimiya) ABC is an isosceles triangel with AB = AC.Wedenote the circumcircle of ABC by Γ. Let D be the point such that DAA and DC are tangent to Γ at A and C repsectively. Prove that DBC ≤ 30◦.

Crux 1963.S21.178*. (K.R.S.Sastry) In triangle ABC, one pair of trisectors of the angles B and C meet at the orthocenter. Show that the other pair of trisectors of these angles meet at the circumcenter.

Crux 1965.S21.207*. (Ji Chen) Let P be a point in the interior of triangle ABC,andlet the lines AP , BP, CP intersect the opposite sides at D, E, F respectively. (a) Prove or disprove that R3 PD· PE · PF ≤ , 8 where R is the circumradius of triangle ABC. Equality holds when ABC is equilateral and P is its center. (b) Prove or disprove that 1 PE · PF + PF · PD+ PD· PE ≤ max(a2,b2,c2), 4 YIU : Problems in Elementary Geometry 351 where a, b,c are the sides of the triangle. Equality holds when ABC is equilateral and P is its center, and also when P is the midpoint o the longest side of triangle ABC.

Crux 1967.S21.209*. (C.J.Bradley) ABC is a triangle and P is a point in its plane. The lines through P parallel to the medians of the triangle meet the opposite sides in points U, V , W . Describe the set of points for which U, V , W are collinear.

Crux 1971.S21.240*. (T.Seimiya) A convex quadrilateral ABCD with AC = BD is inscribed in a circle with center O,andE is the intersection of the diagonals AC and BD.Let P be an interior point of ABCD such that

PAB + PCB = PBC + PDC =90◦.

Prove that O, P , E are collinear.

Crux 1973.S21.245. (K.R.S.Sastry)√ Triangle√ ABC is inscribed in a circle. The chord AD bisects BAC. Assume that AB = 2BC = 2AD. Determine the angles of triangle ABC.

Crux 1977.S21.250*. (D.J. Smeenk) Triangle ABC has circumcenter O.Let be the line through O parallel to BC,andletP be a variable point on . RThe projections of P on BC, CA, AB are Q, R, S respectively. Show that the circle passing through Q, R,andS passes through a ﬁxed point, independent of P .

Crux 1980.S21.279*. (I.Beck and N.Nejlegaard) Find all sets of four points in the plane so that the sum of the distances from each of the points to the other three is a constant.

Crux 1981.S21.255* (T.Seimiya) ABC is an obtuse triangle with A>90◦.LetI and O be the incenter and the circumcenter of ABC. Suppose that [IBC]=[OBC], where [XY Z] denotes the area of triangle XY Z.Provethat

[IAB]+[IOC]=[ICA]+[IBO].

Crux 1983.S21.257*. (K.R.S.Sastry) A convex quadrilateral ABCD has an inscribed circle with center I and also has a circumcircle. Let the line parallel to AB through I meet AD in A and BC in B. Prove that the length of AB is a quarter of the perimeter of ABCD. YIU : Problems in Elementary Geometry 352

Crux 1985 See Crux 2355.

Crux 1987.S21.283*;22(6)276. (H.G—’ulicher) In the ﬁgure, B2C1//A1A2, B3C2//A2A3, and B1C3//A3A1.ProvethatB2C1, B3C2 and B1C2 are concurrent if and only if A C A C A C 1 3 · 2 1 · 3 2 =1. C3B3 C1B1 C2B2

Crux 1991.S21.289. (T.Seimiya) Ω is a fixed circle with center O.LetM be the foot of the perpendicular from O toafixedline,andletP be a varible point on Ω. Let Γ be the circle with diameter PM, intersecting Ω and again at X and Y respectively. Prove that the line XY always passes through a fixed point.

Crux 1993.S21.308*. (W.Pompe) ABCD is a convex quadrilateral inscribed in a circle Γ. Assume that A, B and Γ are ﬁxed and C, D are variable, so that the length of the segment CD is constant. X, Y are the points on the rays AC and BC respectively, such that AX = AD and BY = BD. Prove that the distance between X and Y remains constant. See also Crux 1902. YIU : Problems in Elementary Geometry 353

Crux 1997.S21(9)317*. (C.J.Bradley) ABC is a triangle which is not equilateral, with circumcenter O and orthocenter H.PointK lies on OH so that O is the midpoint of HK. AK meets BC in X,andY , Z are the feet of the perpendiculars from X onto the sides AC, AB respectively. Prove that AX, BY , CA are concurrent or parallel.

Crux 1999.S21(9)320*. (R.M´qrquez) Let ABC be a variable isosceles triangle with constant side a = b and variable side c. Denote the median, angle bisector, and altitude, measured from A to the opposite side, by m, w,andh respectively. Find m − h lim c → a . w − h

Crux 2001.S21.345*. (T.Seimiya) Three similar triangle DBC, ECA, FAB are drawn outwardly on the sides of triangle ABC, such that DBC = ECA = FAB and DCB = EAC = FBA.LetP be the intersection of BE and CF, Q that of AD and BE,andR that of AD and BE.Provethat QR RP PQ = = . AD BE CF

Crux 2008.S22.40*. (J.H.Huang) Let I be the incenter of triangle ABC, and suppose there is a circle with center I which is tangent to each of the excircles of ABC.Provethat ABC is equilateral. Solution. (partial) If this circle touches each of the excircles externally, then it must be the nine-point circle. Now, since the nine-point coincides with the incenter, the triangle must be equilateral.

Crux 2010.S22.40*. (E.Kuczma) In triangle ABC with C =2 A, line CD is the internal angle bisector with D on AB,LetS be the center of the circle tangent to line CA (produced beyond A) and externally tangent to the circumcircles of triangles ACD and BCD.Provethat CS ⊥ AB.

Crux 2011.S22.80. (T.Seimiya) ABC is a triangle with incenter I. BI and CI meet AC and AB at D and E respectively. P is the foot of the perpendicualr from I to DE,andIP meets BC at Q. Suppose that IQ =2IP. Find angle A.

Crux 2012.S22.43*. (K.R.S.Sastry) The number of primitive Pythagorean triangle with a ﬁxed inradius is always a power of 2. Proof. A primitive Pythagorean triangle (m2 − n2, 2mn, m2 + n2) has inradius r = n(m − n). If a given integer r has exactly k distinct prime divisors, then there are 2k ways of factoring r in the form n(m − n)withn and m − n relatively prime. Each of these factorizations gives relatively prime integers m>nleading a primitive Pythagorean triangle. The only r as the inradius of a unique primitive Pythagorean triangle is r =1,andthe triangle has sides 3,4,5.

Crux 2013.S22.44. (W.Pompe) Given a convex n−gon A1A2 ···An (n ≥ 3) and a point P in its plane. Assume that the feet of the perpendiculars from P to the lines A1A2, A2A3 , ..., AnA1 all lie on a circle with center O. (a) Prove that if P belongs to the interior of the n−gon, then so does O. (b) Is the converse to (a) true ? (c) Is (a) still valid for nonconvex n−gons ?

Crux 2015.S22(1)47,125*;24(5)305*. (S.C. Shi and Ji Chen) Prove that √ 1 1 1 27 3 (sin A +sinB +sinC)( + + ) ≥ , A B C π where A, B, C are the angles of a triangle measured in radians.

Crux 2017.S22.82. (D.J. Smeenk) We are given a ﬁxed circle κ and two ﬁxed points A YIU : Problems in Elementary Geometry 355 and B not lying on κ. A variable circle through A and B intersects κ in C and D.Show that the ratio AC · AD BC · BD is constant.

Crux 2019.S22.85*.(P.Penning) In a plane are given a circle C with a diameter l and a point P within C but not on l. Construct the equilateral triangles that have one vertex at P , one on C,andoneonl.

Crux 2021.S22.87. (T.Seimiya) P is a variable interior point of triangle ABC,andAP , BP, CP meet BC, CA, AB at D, E, F respectively. Find the locus of P so that 1 [PAF]+[PBD]+[PCE]= [ABC], 2 where [XY Z] denotes the area of triangle XY Z.

Crux 2024.S22.93. (M.S.Klamkin) It is a known result that if P is any point on the circumcircle of a given triangle ABC with orthocenter H,thenPA2 + PB2 + PC2 − PH2 is a constant. Generalize this result to an n−dimensional simplex.

Crux 2027. (D.J. Smeenk) Quadrilateral ABCD is inscribed in a circle Γ, and has an incircle as well. EF is a diameter of Γ with EF ⊥ BD. BD intersects EF in M and AC in S. Show that AS : SC = EM : MF.

Crux 2029.S22.129*. (J.H.Chen) ABC is a triangle with area F and internal angle bisectors wa, wb, wc. Prove or disprove that √ wbwc + wcwa + wawb ≥ 3 3F.

Crux 2031.S22(3)135. (T.Seimiya) Suppose that α, β, γ are acute angles such that

sin(α − β) sin(β − γ) sin(γ − α) + + =0. sin(α + β) sin(β + γ) sin(γ + α) Provethatatleasttwoofα, β, γ are equal.

Crux 2033.S22(3)137*. (K.R.S.Sastry) The sides AB, BC, CD., DA of a convex quadrilateral ABCD are extended in that order to th points P , Q, R, S such that BP = CQ = DR = AS.IfPQRS is a square, prove that ABCD is also a square. YIU : Problems in Elementary Geometry 356

Crux 2035.S22(4)172. (V.Konecny) If the locus of a point E in an ellipse with ﬁxed foci F and G, prove that the locus of the incenter of triangle EFG is another ellipse.

Crux 2039.S22(3)144. (D.Zhou) Prove or disprove that √ sin A sin B sin C 9 3 + + ≥ . B C A 2π

Crux 2041.S22(4)173. (T.Seimiya) P is an interior point of triangle ABC. AP , BP, CP meet BC, CA, AB at D, E, F respectively. Let M and N be points on segments BF and CE respectively so that BM : MF = EN : NC.LetMN meet BE and CF at X and Y respectively. Prove that MX : YN = BD : DC.

Crux 2043.S22(4)176*. (A.A.Yagubyants) What is the locus of a point interior to a ﬁxed triangle that moves so that the sum of its distances to the sides of the triangle remains constant?

Crux 2047.S22(4)181*. (D.J. Smeenk) ABC is a nn-equilateral triangle with circumcentr O and incenter I. D is the foot of the altitude from A to BC. Suppose that the circumradius R equals the radius ra of the excircle to BC. Show that O, I, D are collinear.

Crux 2051.S22(4)186. (T.Seimiya) A convex quadrilateral ABCD is inscribed in a circle Γ with center O. P is an interior point of ABCD.LetO1, O2, O3, O4 be the circumcenters of triangles PAB, PBC, PCD, PDA respectively. Prove that the midpoints of O1O3, O2O4 and OP are collinear.

Crux 2053. S22(4)187. (J.Kotani) A ﬁgure consisting of two equal and externally tangent circles is inscribed in an ellipse. Find the eccentricity of the ellipse of minimum area.

Crux 2055.S22(4)189. (H.G¨ulicher) In triangle ABC lete D be the point on the ray BC and E on CA such that BD = CE = AB,let be the line through D parallel to AB.IFM is the intersection of and BE,andF that of CM and AB,provethat BA3 = AE · BF · CD.

Crux 2057.S22(4)190. (J.Ciach) Let P be a point inside an equilateral triangle ABC,and let Ra, Rb, Rc and ra, rb, rc denote the distances of P from the vertices and edges respectively of the triangle. Prove or disprove that r r r 27 (1 + a )(1 + b )(1 + c ) ≥ . Ra Rb Rc 8 YIU : Problems in Elementary Geometry 357

Equality holds if P is the center of the triangle. See also Crux 2073.

Crux 2061.S22(5)230 (T.Seimiya) ABC is a triangle with centroid G,andP is a variable interior point of ABC.LetD, E, F be points on sides BC, CA, AB respectively such that PD//AG, P E//BG and P F//CG.Frovethat[PAF]+[PBD]+[PCE] is constant, where [XY Z] denotes the area of triangle XY Z.

Crux 2063.S22(5)233*. (A.A.Yagubyants) Triangle ABC has a right angle at C. (a) Prove that the three ellipses having foci at two vertices of the given triangle, while passing through the third, all share a common point. (b) Prove that the principal vertices of the ellipses of part (a), that is the points where an ellipse meets the axis through its foci) form two pairs of collinear triples. See also Crux 728.S9.116*.

Crux 2067.S22(6)277*. (M.Stupel and V.Oxman) Triangle ABC is inscribed in a circle Γ. Let AA1, BB1, CC1 be the bisectors of angles A, B, C,withA1, B1, C1 on Γ. Prove that the perimeter of the triangle is equal to A B C AA cos + BB cos + CC cos . 1 2 1 2 1 2

Crux 2069.S22(6)278*. (D.J. Smeenk) M is a variable point of side BC of triangle ABC. A line through M intersects the line AB in K and AC in L so that M is the midpoint of the segment KL.PointK is such that ALKK is a parallelogram. Determine the locus of K as M moves on segment BC.

Crux 2071.S22(6)281. (T.Seimiya) P is an interior point of an equilateral triangle ABC so that PB = PC,andBP and CP meet AC and AB at D and E respectively. Suppose that PB : PC = AD : AE. Find angle BPC.

Crux 2073.S22(6)282. (J.Ciach) Let P be an interior point of an equilateral triangle A1A2A3,andletR1 = PA1, R2 = PA2, R3 = PA3. Prove or disprove that 9 R R R ≤ R3. 1 2 3 8 Equality holds if P is the midpoint of a side. See also Crux 1895 and 2057. YIU : Problems in Elementary Geometry 358

Crux 2075.S22(6)286. (C.J.Bradley) ABC is a triangle with A

Crux 2079.S22(7)325. (C.Sanchez-Rubio and I.B.Penyagolosa) An ellipse is inscribed in a rectangle. Prove that the contact points of the ellipse with the sides of the reectangle lie on the rectangular hyperbola which passes through the foci of the ellipse and whose asymptotes are parallel to the sides of the rectangle.

Crux 2082.S22(7)328. (T.Seimiya) ABC is a triangle with A>90◦,andAD, BE and CF are its altitudes (with D on BC,etc.).LetE and F be the feet of the perpendiculars from E and F to BC. Suppose that 2EF =2AD + BC.Find A.

Crux 2084.S22(7)330. (M.S.Klamkin) Prove that B C A B C cos cos ≥ 1 − 2cos cos cos , 2 2 2 2 2 where A, B, C are the angles of a triangle.

Crux 2089.S22(8)366. (M.A.Covas) Let ABCD be a trapezoid with AB CD and let X be a point on segment AB.PutP = CB ∩ AD, Y = CD ∩ PX, R = AY ∩ BD, and T = PR∩ AB.Provethat 1 1 1 = + . AT AX AB

Crux 2091.S22(8)367*. (T.Seimiya) Four points A, B, C, D are on a line in this order. We put AB = a, BC = b, CD = c. Equilateral triangles ABP , BCQ and CDR are constructed on the same side of the line. Suppose that PQR = 120◦. Find the relation between a, b and c.

Crux 2093.S22(8)371. (W.Janous) LetA, B, C be the angles (in radians) of a triangle. Prove or disprove √ 1 1 1 27 3 (sin A +sinB +sinC)( + + ) ≤ . π − A π − B π − C 4π

Crux 2096.S22(8)374. (D.J. Smeenk) Triangle A1A2A3 has circumcircle Γ. The tangents at A1, A2, A3 to Γ intersect (the extensions of) A2A3, A3A2, A1A2 respectively in B1, B2, B3. YIU : Problems in Elementary Geometry 359

The second tangent to Γ through B1, B2, B3 touch Γ at C1, C2, C3 respectively. Show that A1C1, A2C2, A3C3 are concurrent.

Crux 2101.S23(1)49. (J.Chen) Prove that for any k ≤ 1,

ak 3 ≥ ak, A π where the sums are cyclic. [The case k = 1 is known; see item 4.11 (p.170) of Mitrinovic et. al].

Crux 2102.S23(1)49. (T.Seimiya) ABC is a triangle with incenter I.LetP and Q be the feet of the perpendiculars from A to BI and CI respectively. Prove that AP AQ A + =cot . BI CI 2

Crux 2103.S23(1).52. (T.Seimiya) ABC is a triangle. Let D be the point on side BC produced beyond B such that BD = BA,andletM be the midpoint of AC. The bisector of ABC meets DM at P .Provethat BAP = ACB.

Crux 2106.S23(1).55. (YANG Kechang) A quadrilateral has sides a, b, c, d (in that order) and area F .Provethat

2a2 +5b2 +8c2 − d2 ≥ 4F.

When does equality hold? YIU : Problems in Elementary Geometry 360

Crux 2107.S23(1)57*. (D.J. Smeenk) Triangle ABC is not isosceles nor equilateral, and has sides a, b, c. D1 and E1 are points of BA and CA or their productions so that BD1 = CE1 = a. D2 and E2 aere points of CB and AB or their productions so hat CD2 = AE2 = b. Show that D1E1//D2E2.

Crux 2109.S23(1)60*. (V.Oxman) In the plane are given a triangle and a circle passing through two of the vertices of the triangle and also through the incenter of the triangle. (The incenter and the center of the circle are not given). Construct, using only an unmarked ruler, the incenter. Remark. The “easy” construction fails if the triangle is isosceles, with the circle passing through the base vertices. In that case, the incenter is the midpoint of the arc, and cannot be constructed with an unmarked ruler.

Crux 2114.S23(2)114. (T.Seimiya) ABCD is a square with incircle Γ. A tangent to Γ meets the sides AB and AD and the diagonal AC at P , Q,andR respectively. Prove that AP AR AQ + + =1. PB RC QD

Crux 2116.S23(2)116. (YANG Kechang) A triangle has sides a, b, c and area F .Prove that √ 25 5(2F )6 a3b4c5 ≥ . 27 When does equality hold?

Crux 2117.S23(2)116. (T.Seimiya) ABC is a triangle with AB > AC, and the bisector of A meets BC at D.LetP be an interior point of the side AC.Provethat BPD < DPC.

Crux 2120.S23(2)122. (M.E.Kuczma) Let A1A3A5 and A2A4A6 be nondegenerate triangles in the plane. For i =1,...,6, let i be the perpendicular from Ai to the line Ai−1Ai+1 (where, of course, A0 = A6 and A7 = A1). If 1, 3, 5 concur, prove that 2, 4, 6 also concur.

Crux 2124.S23(3)171*. (C.Shevlin) Suppose that ABCD is a quadrilateral with CDB = CBD =50◦ and CAB = ABD = BCD.ProvethatAD ⊥ BC.

Crux 2127.S23(3)177. (T.Seimiya) ABC is an acute triangle with circumcenter O,and D is a point on the minor arc AC of the circumcircle D = A, C). Let P be a point on the side AB such that ADP = OBC,andletQ be a point onth side BC such that CDQ = OBA. Prove that DPQ = DOC and DQP = DOA. YIU : Problems in Elementary Geometry 361

Crux 2128.S23(3)178. (T.seimiya) ABCD is a square. Let P and Q be interior points onthe sides BC and CD respectively, and let E and F be the intersetions of PQ with AB and AD respectively. Prove that 5 π ≤ PAQ+ ECF < π. 4

Crux 2130.S23(3)179*. (D.J. Smeenk) A and B are fixed points, and is a fixed line passing through A. C is a variable point on staying on one side of A. The incircle of ABC touches BC at D an AC at E. Show that the line DE passes through a fixed point. Solution. Set upan oblique coordinate system with A as origin and the lines l and AB as axes. Suppose the segments AB, AE,andEC have lengths c, x,andy respectively. Then the points B, E,andC have coordinates (0,c), (x, 0), and (x + y,0) respectively. If the incircle touches AB at the point F , then it is easy to see that AF = AE = x,sothat

BD = BF = c − x.

Note also that CD = CE = y. It follows that the point D has coordinates 1 1 [y(0,c)+(c − x)(x + y,0)] = (c − x)(x + y),cy . c − x + y c − x + y

A typical point on the line DE has coordinates tD +(1− t)E for some t(= −1). Explicitly, this is the point ty ty x + · (c − 2x), · c . c − x + y c − x + y ty 1 1 By choosing t such that c−x+y = 2 , we obtain the point 2 (c, c) independent of x and y.This ﬁxed point can be identiﬁed as the midpoint of the segment BB,whereB =(c, 0) is the point on l such that AB = AB = c.

Having located the ﬁxed point by coordinate geometry, we can now give a simple synthetic proof.

Alternative Solution Let B be the point on l such that AB = AB = c. Suppose AE = x and EC = y (as above). Regarding DE as a transversal of the triangle BBC intersecting BB at the point P ,wehaveCD = y, DB = c − x, BE = AB − AE = c − x,andEC = y.By Menelau’s theorem, BP BE CD · · = −1. PB EC DB Here, BE − c−x (i) EC = y , E dividing B C externally, and YIU : Problems in Elementary Geometry 362

CD y (ii) DB = c−x . BP It follows that PB =1,andDE passes through the midpoint of the segment BB .

Crux 2133.S23(4)249. (K.R.S.Sastry) Similar non-square rectangels are placed outwardly on the sides of a parallelogram π. Prove that the centers of these rectangles also form a non-square rectangle if and only if π is a non-square rhombus.

Crux 2136.S23(3)185. (G.P.Henderson) Let a, b, c be the lengths of the sides of a triangle. Given the values of p = a and q = ab,provethatr = abc can be estimated with 1 an error of at most 26 r.

Crux 2137.(corrected 22(7))S23(3)187*.. (A.A.Yagubyants) Three circles of (equal) radius t passes through a point T , andareeach inside triagle ABC and tangent t two of its sides. Prove that 2R (i) t = R+2 ; (ii) T lies on the line segment joining the centers of the circumcircle and the incircle of ABC. See also Crux 694.

Crux 2138.S23(3)188. (C.J.Bradley) ABC is an acute angle triangle with circumcenter O. AO meets the circle BOC again at A, BO meets the circle COA again at B,andCO meets the circle AOB again at C.Provethat[ABC] ≥ 4[ABC], where [XY Z] denotes the area of triangle XY Z.

Crux 2139.(corrected 22(5))S23(3)190. (W.Pompe) Point P lies inside triangle ABC. Let D, E, F be the orthogonal projections from P onto the lines BC, CA, AB respectively. Let O,andR denote the circumcenter and circumradius of the triangle DEF respectively. Prove that √ [ABC] ≥ 3 3R R2 − OP 2, where [XY Z] denotes the area of triangle XY Z.

Crux 2141.S23(4)250. (T.Seimiya) A1A2A3A4 is a quadrilateral. Let B1, B2, B3, B4 be points on the sides A1A2, A2A3, A3A4, A4A1 respectively, such that

A1B1 : B1A2 = A4B3 : B3A3 and A2B2 : B2A3 = A1B4 : B4A4. YIU : Problems in Elementary Geometry 363

Let P1, P2, P3 and P4 be points on B4B1, B1B2, B2B3,andB3B4 such that

P1P2//A1A2,P2P3//A2A3,P3P4//A3A4.

Prove that P4P1//A4A1.

Crux 2142.S23(4)252. (V.Oxman) In the plane are given an arbitrary quadrangle and bisectors of three of its angles. Construct, using only an unmarked ruler, the bisector of the fourth angle.

Crux 2146.S23(5)303. (T.Seimiya) ABC is a triangle with AB > AC, and the bisector of A meets BC at D.LetP be an interior point on the segment AD,andletQ and R be the points of intersection of BP and CP with sides AC and AB respectively. Prove that PB − PC > RB − QC > 0.

Crux 2148.S23(5)306*. (A.A.Yagubyants) Suppose that AD, BE and CF are the altitudes of triangle ABC. Suppoe that L, M, N are points on BC, CA, AB respectively, such that BL = DC, CM = EA, AF = NB.Provethat 1. the perpendiculars to BC, CA, AB at L, M, N respectively are concurrent; 2. the point of concurrency lies on the Euler line of triangle ABC.

Crux 2149.S23(5)306*. (J.B.Romero M´arquez) Let ABCD be a convex quadrilateral and O the point of intersection of the diagonals AC and BD.LetABCD be the quadrilateral whose vertices A, B, C, D are the feet of the perpendiculars drawn from the point O to the sides BC, CD, DA, AB respectively. Prove that ABCD is an inscribed (cyclic) quadrilateral if and only if ABCD is a circumscribing quadrilateral (AB, BC, CD, DA are tangents to a circle).

Crux 2151.S23(5)310*. (T.Seimiya) ABC is a triangle with B =2 C.LetH be the foot of the perpendicular from A to BC,andletD be the point on the side BC where the excircle touchese BC.ProvethatAC =2HD.

Crux 2154.S23(5)314*. (K.R.S.Sastry) In a convexpentagon, the medians are concurrent. If the concurrence point sections each median in the same ratio, ﬁnd its numerical value. (A median of a pentagon is the line segment between a vertex and the midpoint of the third side from the vertex).

Crux 2156.S23(5)318*. (H.T.Wee) ABCD is a convex quadrilateral with perpedicular diagonals AC and BD. S and Y are points in the interior of sides BC and AD resepectively YIU : Problems in Elementary Geometry 364 such that BX BD DY = = . CX AC AY BC·XY Evaluate BX·AC .

Crux 2160.S23(6)370. (T.Seimiya) ABC is a triangle with A<90◦.LetP be an interior point of ABC such that BAP = ACP and CAP = ABP .LetM and N be the incenters of ABP and ACP respectively, and let R1 be the circumradius of AMN.Prove that 1 1 1 1 = + + . R1 AB AC AP

Crux 2162.S23(6)373*. (D.J. Smeenk) In ABC, the Cevian lines AD, BE, CF concur at P .[XY Z] is the area of XY Z.PRovethat

[DEF] PD PE PF = · · . 2[ABC] PA PB PC

Crux 2164.S23(6)377*. (T.Seimiya) Let D be a point on the side BC of triagle ABC, and let E, F be the incenters of triangle ABD and ACD respectively. Suppose that B, C, E, F are concyclic. Prove that AD + BD AB = . AD + CD AC

Crux 2165.S23(6)378. (H.T.Wee) Given a triangle ABC, prove that there exists a unique pair of points P and Q such that the triangles ABC, PQC and PBQ are directly similar; that is, ABC = PQC = PBQ and BAC = QP C = BPQ, and the three similar triangle have the same orientation. Find a euclidean construction for the points P and Q.

Crux 2166.S23(6)380*. (K.R.S.Sastry) In a right - angled triangle, establish the existence of a unique interior point with the property that the line through the point perpendicular to any side cuts oﬀ a triangle of the same area.

Crux 2169.S23(7)434. (D.J. Smeenk) AB is a ﬁxed diameter of circle C1 := ◦OR. P is an arbitrary point on its circumference. Q is the projection onto AB of P .CircleP1PQ intersects C1 at C and D. CD intersects PQ at E. F is the midpoint of AQ. FG ⊥ CD,where G ∈ CD. Show that (1) EP = EQ = EG; (2) A, G and P are collinear. YIU : Problems in Elementary Geometry 365

Crux 2171.S23(7)436*. (J.B. Romero M´arquez) Let P be an arbitrary point taken on an ellipse with foci F1 and F2, and directrices d1 and d2 respectively. Draw the straight line through P which is parallel to the major axis of the ellipse. This lien intersects d1 and d2 at points M and N respectively. Let P be the point where MF1 intersects NF2. Prove that the quadrilateral PF1P F2 is cyclci. Does the result also hold in the case of a hyperbola?

Crux 2177.S23(7)464. (T.Seimiya) ABCD is a convex quadrilateral, with P the intersection of its diagonals and M the midpoint of AD. MP meets BC at E. Suppose that BE : EC = AB2 : CD2. Characterize quadrilateral ABCD.

Crux 2178.S23(7)447*. (C.J.Bradley) If A, B, C are the angles of a triangle, prove that

sin A sin B sin C ≤ 8(sin3 A cos B cos C +sin3 B cos C cos A +sin√ 3 C cos A cos B ≤ 3 3(cos2 A +cos2 B +cos2 C).

Crux 2183.S23(8)513. (V.Konecny) Suppose A, B, C are the angles of a triangle, and that k, , m ≥ 1. Show that

0 < sink A sin; B sinm C ≤ kk;mmSS/2(Sk2 + P )−k/2(S2 + P )−;/2(Sm2 + P )−m/2, where S = k + + m and P = km.

Crux 2186.S23(8)519*. (V.N.Murty) 1 GI2 = (a − b)(b − c)(b + c − a). 9(a + b + c) Deduce from this 1 GI2 = (s2 +5r2 − 16Rr). 9

Crux 2188.S23(8)522. (V.Oxman) Suppose that a, b, c are the sides of a triangle with semiperimeter s and area .Provethat 1 1 1 s + + < . a b c Solution. Note that a =(s − b)+(s − c), and 1 1 1 s(s − a) 2s − a = ≤ = < . a (s − b)+(s − c) 2 (s − b)(s − c) 2 4 YIU : Problems in Elementary Geometry 366

Here, equality does not hold since s = s − a. Similarly, 1 2s − b 1 2s − c < and < . b 4 c 4 It follows that 1 1 1 (2s − a)+(2s − b)+(2s − c) 4s s + + < = = . a b c 4 4

Crux 2189.S23(8)523. (T.Seimiya) The incircle of a triangle ABC touches BC at D.LetP and Q be variable points on sides AB and AC respectively such that PQ is tangent to the incircle. Prove that the area of triangle DPQ is a constant multiple of BP · CQ.

Crux 2190.S23(8)525*. (W.Janous) Determine the range of

sin2 A sin2 B sin2 C + + A B C where A, B, C are the angles of a triangle.

Crux 2194.S23(8)530*. (C.J.Bradley) Prove or disprove that it is possible to ﬁnd a triangle ABC and a transversal NML with N lying between A and B, M lying between A and C,andL lying on BC produced, such that BC, CA, AB, NB, MC, NM, ML,andCL are all of integer length, and NMCB is a cyclic inscriptible quadrilateral.

Crux 2198.S24(1)49*. (V.N.Murty) Prove that if a, b, c are the lengths of the sides of a triangle, 2 1 2 1 2 1 (b − c)2( − )+(c − a)2( − )+(a − b)2( − ) ≥ 0, bc a2 ca b2 ab c2 with equality if and only if a = b = c.

Crux 2201.S24(1)53. (T.Seimiya) ABCD is a convex quadrilateral, and O is the intersection of its diagonals. Let L, M, N be the midpoints of DB, BC, CA respectively. Suppose that AL, OM, DN are concurrent. Show that either AD//BC or [ABCD]=2[OBC], where [F] denotes the area of ﬁgure F.

Crux 2202.S24(1)54*. (W.Janous) Supoose that n ≥ 3. Let A1 ···An be a convex n−gon. Determine the greatest constant Cn such that n 1 n 1 ≥ C . A n π − A k=1 k k=1 k YIU : Problems in Elementary Geometry 367

Determine when equality occurs.

Crux 2203.S24(1)56. (W.Janous) Let ABCD be a quadrilateral with incircle I.Dentoe by P , Q, R and S the points of tangency of sides AB, BC, CD and DA, respectively with I. Determine all possible values of (PR,QS) such that ABCD is cyclic. √ Crux 2204.S24(1)57. (S.Arslanaglf) For triangle ABC such that R(a + b)=c ab,prove that 3 r< a. 10

Crux 2205.S24(1)58*. (V.Koneˇcn´y) Find the least positive integer n such that the expression sinn+2 A sinn+1 B sinn C has a maximum which is a rational number. Here, A, B, C are the angles of a variable triangle.

Crux 2209.S24(2)112*. (M.A.Caberz´on Ochoa) Let ABCD be a cyclic quadrilateral having perpendicular diagonals crossing at P . Project P onto the sides of the quadrilateral. 1. Prove that the quadrilateral obtained by joining these four projections is inscribable and circumscribable. 2. Prove that the circle which passes through these four projections also passes through the midpoints of the sides of the given quadrilateral.

Crux 2215.S24(2)121. (T.Chronis) Let P be a point inside a triangle ABC. Determine P such that PA+ PB+ PC is a maximum. See also CMJ346.872.S892. There it is shown that the maximum occurs at a vertex where two longest sides of the triangle meet. There is no maximum inside the triangle.

Crux 2224.S24(3)184. (W.Pompe) Point P lies inside triangle ABC. Triangle BCD is erected outwardly on side BC such that BCD = ACP and CBD = ABC.Provethatif the area of quadrilateral PBDC is equal to the area of triangle ABC, then triangle ACP and BCD are similar.

Crux 2230.S24(3)191. (W.Pompe) Triangels BCD and ACE are constructed outwardly on sides BC and CA of triangle ABC such that AE = BD and BDC + AEC = 180◦.The point F is chosen to lie on the segment AB so that AF DC = . FB CE YIU : Problems in Elementary Geometry 368

Prove that DE EF FD = = . CD + CE BC AC

Crux 2231.S24(4)242. (H.G¨ulicher) In quadrilateral P1P2P3P4, suppose that the diagonals intersect at the point M = Pi,(i =1, 2, 3, 4). Let MP1P4 = α1, MP3P4 = α2, MP1P2 = β1,and MP3P2 = β2.Provethat

|P1M| cot α1 ± cot β1 λ13 := = , |MP3| cot α2 ± cot β2 where the +(−) sign holds if the line segment PqP3 is located inside (outside) the quadrilateral.

Crux 2234(=2287).S24(4)247;(8)525. (V.Oxman) Given triangle ABC, its centroid G and its incenter I, construct, using only an unmarked ruler, its orthocenter H.

◦ Crux 2235.S24(4)249. (D.J. Smeenk) Triangle ABC has angle CAB =90 .LetΓ1(O, R) be the cirucmcircle and Γ2(T,r) be the incircle. The tangent to Γ1 at A and the polar line of A with respect to Γ2 intersect at S. The distance from S to AC and AB are denoted by d1 and d2 respectively. Show that (a) ST//BC, (b) |d1 − d2| = r.

Crux 2236.S24(4)250*. (V.Oxman) Let ABC be an arbitrary triangle and let P be an arbitrary point in the interior of the circumcircle of triangle ABC,LetK, L, M denote the feet of the perpendiculars from P to the lines AB, BC, CA respectively. Prove that 1 [KLM] ≤ [ABC]. 4

Crux 2237.S24(4)251*. (M.D.Visiliou) ABCD is a square with incircle Γ. Let be a tangent to Γ. Let A, B, C, D be points on such that AA, BBm CC, DD are all perpendicular to .ProvethatAA · CC = BB · DD.

Crux 2240.S24(5)312. (V.Oxman) Let ABC be an arbitrary triangle with the points D, BD ≤ BF ≤ AE ≤ AF E, F on the sides BC, CA, AB respectively. so that DC FA 1and EC FB.Provethat 1 [DEF] ≤ [ABC] 4 with equality if and only if two of the three points D, E, F (atleast)aremidpointsofthe corresponding sides. YIU : Problems in Elementary Geometry 369

Crux 2241.S24(5)313*. (T.Seimiya) Triangle ABC (AB = AC) has incenter I and circumcenter O. The incircle touches BC at D. Suppose IO ⊥ AD.ProvethatAD is a symmedian of triangle ABC. (A symmedian is the reﬂection of the median in the internal angle bisector).

Crux 2242.S24(5)314*. (K.R.S.Sastry) ABCD is a parallelogram. A point P lies in the plane such that (1) the line through P parallel to DA meets DC at K and AB at L, (2) the line through P parallel to AB meets AD at M and BC at N,and (3) the angle between KM and LN is equal to the non-obtuse angle of the parallelogram. Find the locus of P .

Crux 2244.S24(5)317. (T.Seimiya) ABC is a triangle and D is a point on AB produced beyond B such that BD = AC,andE is a point on AC produced beyond C such that CE = AB. The perpendicular bisector of BC meets DE at P .Provethat BPC = BAC.

Crux 2246.S24(5)318*. (D.J. Smeenk) Suppose that G, I and O are the centroid, the incenter, and the circumcenter of a non-equilateral triangle ABC. The line through B, perpendicular to OI intersects the bisector of angle BAC at P . The line through P ,parallelto AC intersects BC at M. Show htat I, G, M are collinear.

Crux 2250.S24(6)372. (T.Seimiya) ABC is a scalene triangle with incenter I.LetD, E, F be the points where BC, CA, AB are tangent to the incircle respectively, and let L, M, N be the midpoints of BC, CA, AB respectively. Let , m, n be the lines through D, E, F parallel to IL, IM, IN respectively. Prove that , m, n are concurrent. YIU : Problems in Elementary Geometry 370

Crux 2251(=2288).S24(6)373;(8)525. (V.Oxman) In the plane you are given a circle (but not its center), and points A, K, B, D, C on it, so that arc AK =arcKB and arc BD = arc DC. Construct, using oly an unmarked straightedge, the mid-point of arc AC.

Crux 2252.S24(6)375. (K.R.S.Sastry) Prove that the nine-point circle of a triangle trisects a median if and only if the side length of the triangle are proportional to its median lengths in some order.

Crux 2253.S24(6)376*. (T.Seimiya) ABC is a triangle and Ib, Ic are the excentres of ABC relative to the sides CA, AB respectively. Suppose that

2 2 2 2 2 2 2 2 IbA + IbC = BA + BC , and IcA + IcB = CA + CB . Prove that ABC is equilateral.

Crux 2254.S24(6)377*. (T.Seimiya) ABC is an isosceles triangle with AB = AC.Let D be the point on side AC such that CD =2AD.LetP be the point on the segment BD such that AP C =90◦.Provethat ABP = PCB.

Crux 2255.S24(6)378;25(2)113*. (T.Seimiya) Let P be an arbitrary point of an equilateral triangle ABC.Provethat

| PAB − PAC|≥| PBC − PCB|.

Crux 2257.S24(7)427. (W.Pompe) The diagonals AC and BD of a convex quadrilateral ABCD intersect at the point O.Let OK, OL, OM, ON be the altitudes of triangles ABO, BCO4,CDO4, DAO respectively. Prove that if OK = OM and OL = ON,thenABCD is a parallelogram.

◦ Crux 2258.S24(6)380*. (W.Pompe) In√ a right - angled triangle ABC (with C =90 ),D lies√ on the segment BC so that BD = AC 3. E lies on the segment AC and satisﬁes AE = CD 3. Find the angle between AD and BE.

Crux 2259.S24(8)509*. (Yiu) Let X, Y, Z be the projections of the incenter of ABC on the sides BC,CA and AB respectively. Let X,Y,Z be the points on the incircle diametrically opposite to X, Y, Z respectively. Show that the lines AX,BY,CZ are concurrent.

Crux 2262.S24(7)431*. (Juan-Bosco Romero M´qrquez) Consider two triangles ABC and ABC such that A ≥ 90circ and A ≥ 90◦, and whose sides satisfy a>b≥ c and ≥ a >b c . Denote the altitude to sides a and a by ha and ha.Provethat YIU : Problems in Elementary Geometry 371

1 1 1 (a) ≥ + ; haha bb cc 1 1 1 (b) ≥ + . haha bc b c

Crux 2263.S24(7)432*. (T.Seimiya) ABC is a triangle, and the internal bisectors of B, C, meet AC, AB at D, E respectively. Suppose that BDE =30◦. Characterize ABC.

Crux 2264.S24(8)510. (T.Seimiya) ABC is a right angled triangle with the right angle at A.PointsD and E are on sides AB and AC respectively such that DE//BC.PointsF and G are the feet of the perpendiculars from D, E to BC respectively. Let I, I1, I2, I3 be the incenters of ABC, ADE, BDF, CEG, respectively. Let P be the point such that I2P//I1I3 and I3P//I1I2. Prove that the segment IP is bisected by the line BC.

Crux 2265.S24(7)433*. (W.Pompe) Given triangle ABC,letABX and ACY be two variable triangles constructed outwardly on sides AB and AC of triangle ABC, such that the angles XAB and YAC are ﬁxed, and XBA + YCA= 180circ.Prove that all the lines XY pass through a common point.

Crux 2266.S24(7)434*. (W.Pompe) BCLK is the square constructed outwardly on side BC of an acute triangle ABC.LetCD be the altitude of triangle ABC (with D on AB), and let H be the orthocenter of triangle ABC. If the lines AK and CD meet at P , show that HP AB PD = CD.

Crux 2267.S24(8)511. (C.Kimberling and P.Yﬀ) In the plane of triangle ABC,letF be the Fermat point and F its isogonal conjugate. Prove that the circle through F centered at A, B, C meet pairwise in the vertices of an equilateral triangle having center F .

Crux 2270.S24(7)437*. (D.J. Smeenk) Given triangle ABC with sides a, b,c,acircle, center P , and radius ρ intesects sides BC, A, AB in A1, A2; B − 1, B2; C1, C2 respectively so that A A B B C C 1 2 = 1 2 = 1 2 = λ>0. a b c Determine the locus of P .

Crux 2276.S24(8)514. (D.J. Smeenk) Quadrilateral ABCD is cyclic with circumcircle Γ(O, R). Show that the nine - point circles of triangles BCD, CDA, DAB and ABC have a point in common, and characterize that point. YIU : Problems in Elementary Geometry 372

Crux 2279.S24(8)515*. (W.Janous) With the usual notation for a triangle, prove that sr sin3 A cos B cos C = (2R2 − s2 +(2R + r)2). 4R4 cyclic

In Crux 2178.S23(447), Florian Herzig showed that sin3 A cos B cos C = sin A cos2 A. cyclic

Crux 2280.S24(8)516. (T.Seimiya) ABC is a triangle with incenter I.LetD be the second intersection of AI with the circumcircle of ABC.LetX, Y be the feet of the perpendiculars from I to BD, CD respectively. 1 Suppose that IX + IY = 2 AD.Find BAC.

Crux 2281.S24(8)517. (T.Seimiya) ABC is at triangle, and D a point on the side BC produced beyond C, such that AC = CD.LetP be the second intersection of the circumcircle of triangle ACD with the circle on diameter BC.LetE be the intersection of BP with AC, and let F be the intersection of CP with AB. Prove that D, E, F are collinear.

Crux 2282.S24(8)518*. (D.J. Smeenk) A line intersects the sides BC, CA, AB of triangle ABC at D, E, F respectively such that D is the midpoint of EF. Determine the minimum value of |EF| and express its length as elements of triangle ABC.

Crux 2283.S24(8)519*. (W.Pompe) You are given triangle ABC with C =60circ. Suppose E is an interior point of line egment AC such that CE < BC.SUpposethatD is an interior point of line segment BC such that AE BC = − 1. BD CE Suppose that AD and BE intersect in P , and the circumcircles of AEP and BDP intersect in P and Q.ProvethatQE//BC.

Crux 2284(corrected).S24(8)521*. (T.Seimiya) ABCD is a rhombus with A =60◦. Suppose that E, F are points on the sides AB, AD respectively, and that CE, CF meet BD at P , Q respectively. Suppose that BE2 + DF 2 = EF2.Show that BP23+DQ2 = PQ2.

Crux 2286. Proposed by Toshio Seimiya, Kawasaki, Japan. YIU : Problems in Elementary Geometry 373

ABCD is a rhombus with A =60◦. Suppose that E, F are points on th sides AB, AD respectively, and that CE, CF meet BC at P , Q respectively. Suppose that BE2 +DF 2 = EF2. Prove that BP2 + DQ2 = PQ2.

Crux 2287=2234.S24(4)247.

Crux 2288=2251.S24(6)373.

Crux 2301.S25(1)52. (C.J.Bradley) Suppose that ABC is a triangle with sides a, b, c, that P is a point in the interior of triangle ABC,andthatAP meets the circle BPC again at A. Deﬁne B and C similarly. Prove that the perimeter of the hexagon ABCABC satisﬁes √ √ √ P ≥ 2( ab + bc + ca).

Crux 2302.S25(1)53. (T.Seimiya) Suppose that the bisector of angle A of triangle ABC intersects BC at D. Suppose that AB + AD = CD and AC + AD = BC. Determine the angles B and C.

Crux 2303.S25(1)55. (T.Seimiya) Suppose that ABC is a triangle with angles B and C ◦ 1 satisfying C =90 + 2 B, that the exterior bisector of angle A intersects BC at D, and that the side AB touches the incircle of triangle ABC at E.ProvethatCD =2AE.

Crux 2304.S25(1)56. (T.Seimiya) An acute angled triangle ABC is given, and equilateral triangles ABD and ACE are drawn outwardly on the sides AB and AC. Suppose that CD and BE meet AB and AC at F and G respectively, and that CD and BE intersect at P . Suppose that the area of the quadrilateral AF P G is equal to the area of triangel PBC. Determine angle BAC.

Crux 2309.S25(2)114*. (C.J.Bradley) Suppose that ABC is a triangle and that P is a point on the circumcircle, distinct from A, B, C.Denote by SA the circle with center A and radius AP . Deﬁne SB and SC similarly. Suppose that SA and SB intersect at P and PC . Deﬁne PB and PA similarly. Prove that PA, PB, PC are collinear.

Crux 2314.S25(2)117. (T.Seimiya) Given triangle ABC with AB < AC. The bisectors of angles B and C meet AC and AB at D and E respectively, and DE intersects BC at F . 1 − Suppose that DFC = 2 ( DBC ECB). Determine angle A. YIU : Problems in Elementary Geometry 374

Crux 2316.S25(2)119. (T.Seimiya) Given trinalge ABC with angles B and C satisfying ◦ 1 C =90 + 2 B. Suppose that M is the midpoint of BC, and that the circle with center A and radius AM meets BC again at D.ProvethatMD = AB.

Crux 2318.S25(2)123*. (V.Konecny) Suppoe that ABC is a triangle with circumcenter O and circumradius R. Consider the bisector of any side (say AC)andletP (the pedal point) be any point on inside the circumcircle. Let K, L, M denote the feet of the perpendicualrs from P to the lines AB, BC, CA respectively. Show that the area [KLM] is a decreasing function of OP.

Crux 2319.S25(2)124*. (F.Herzig) Suppose that UV is a diameter of a semicircle, and that P , Q are two points on the semicircle with UP < UQ. The tangents to the semicircle at P and Q meet at R. Suppose that S is the point of intersection of UP and VQ. Prove that RS is perpendicular to UV.

Crux 2320.S25(2)126*. (D.J. Smeenk) Two circles on the same side of the line are tangenttoitatD. The tangnets to the smaller circle from a variable point A on the large circle intersect at B andC.Ifb and c are the radii of the incircles of triangles ABD and ACD,prove that b + c is independent of the choice of A.

Crux 2322.S25(3)175. (K.R.S.Sastry) Suppose that the ellipse E has equation

x2 y2 + =1. a2 b2 Suppose that Γ is any circle concentric with E. Suppose that A is a point on E and B is a point on Γ such that AB is tangent to both E and Γ. Find the maximum length of AB.

Crux 2326.S25(3)178*. (W.Janous) Prove or disprove

2 (1 − sin A )(1 + 2 sin A ) 9 < 2 2 ≤ . π π − A 2π

Crux 2333.S25(3)187. (D.J. Smeenk) YouaregiventhatD and E are points on the sides AC and AB respectively of triangle ABC.Also,DE is not parallel to CB. Suppose that F and G are points on BC and ED respectively such that

BF : FC = EG : GD = BE : CD. YIU : Problems in Elementary Geometry 375

Show that GF is parallel to the angle bisector of BAC.

Crux 2334.S25(3)188. (T.Seimiya) Suppose that ABC is a triangle with incentre I,and that BI, CI meet AC, AB at D, E respectively. Suppose that P is the intersection of AI with DE. Suppose that PD = PI. Find angle ACB.

Crux 2335.S25(3)190*. (T.Seimiya) Triangle ABC has circumcircle Γ. A circle Γ is internally tangent to Γ at P , and touches sides AB, AC at D, E respectively. Let X, Y be the A feet of the perpendiculars from P to BC, DE respectively. Prove that PX = PYsin2 .

Crux 2336.S25(3)191. (T.Seimiya) The bisector of angle A of a triangle ABC meets BC at D.LetΓandΓ be the circumcircles of triangles ABD and ACD respectively, and let P , Q be the intersections of AD with the common tangents to Γ, Γ respectively. Prove that PQ2 = AB · AC.

Crux 2338.S25(4)243*. (Seimiya) Suppose ABCD is a convex cyclic quadrilateral, and P is the intersection of the diagonals AC and BD.LetI1, I2, I3, I4 be the incentres of triangles PAB, PBC, PCD and PDA respectively. Suppose that I1, I2, I3, I4 are concyclic. Prove that ABCD has an incircle.

Crux 2339.S25(5)309*. (T. Seimiya) A rhombus ABCD has incircle Γ, and Γ touches AB at T . A tangent to Γ meets sides AB, AD at P , S respectively, and the line PS meets BC, CD at Q, R respectively. Prove that 1 1 1 (a) PQ + RS = BT , and 1 − 1 1 (b) PS QR = AT .

Crux 2342.S25(4)249*. (D.J. Smeenk) Given A and B are ﬁxed points of circle Γ. The point C movesonΓ,ononesideofAB. D and E are points outside triangle ABC such that triangles ACD and BCE are both equilateral. (a) Show that CD and CE each pass through a ﬁxed point of Γ when C moves on Γ. (b) Determine the locus of the midpoint of DE.

Crux 2346.S25(5)311. (J.B.Romero M´arquez) The angles of triangle ABC satisfy A>B>C. Suppose that H is the foot of the perpendicualr from A to BC,thatD is the foot of the perpendicular form H to AB,thatE is the foot of the perpendicular from H to AC,that P is the foot of the perpendicular from D to BC,andthatQ is the foot of the perpendicular from E to AB.ProvethatA is acute, right, or obtuse according as AH − DP − EQ is positive, zero, or negative. YIU : Problems in Elementary Geometry 376

Crux 2348.S25(5)312*. (D.J. Smeenk) Without the use of trigonometrical formulae, prove that 1 sin 54◦ = +sin18◦. 2

Crux 2349.S25(4)255. (V.Konecny) Suppose that ABC has acute angles such that A sin2 A sin sin(B + ). 2 2 2 2 Solution: Let the bisectors of angles A and B intersect their opposite sides at P and Q respectively. In standard notation, ab ab CP = ,CQ= . b + c a + c By the law of sines,

sin2 A sin B sin(B + A ) 2 sin(B + A ) sin B 2 2 sin A · 2 · 2 2 A B = 2 A B sin B sin 2 sin(A + 2 ) sin B sin 2 sin(A + 2 ) a2 CQ b a CQ a b + c ab + ac = · · = · = · = < 1. b2 a CP b CP b a + c ab + bc since ac < bc. Since all the sines involved are positive, this results holds under the assumption A

Crux 2350.S25(4)255. (C.J.Bradley) Suppose that the centroid of triangle ABC is G, and that M and N are the midpoints of AC and AB respectively. Suppose that circles ANC and AMB meet at (A and) P ,andthatcircleAMN meets AP again at T . (a) Determine AT : AP . (b) Prove that BAG = CAT.

Crux 2352.S25(5)315*. (C.J.Bradley) Determine the shape of ABC if cos A cos B cos(A−B)+cosB cos C cos(B −C)+cosC cos A cos(C −A)+2cosA cos B cos C =1. YIU : Problems in Elementary Geometry 377

Crux 2353.S25(5)316*. (C.J.Bradley) Determine the shape of ABC if sin A sin B sin(A − B)+sinB sin C sin(B − C)+sinC sin A cos(C − A)=0.

Crux 2354(corrected).S25(5)317. (H.G¨ulicher) In triangle P1P2P3, the line joining Pi−1Pi+1 meets a line σj at the point Si,j,(i, j =1, 2, 3, all indices taken modulo 3), such that all the points Si,j, Pk are distinct, and diﬀerent from the vertices of the triangle. (1) Prove that if all the point Si,i are non - collinear, then any two of the following conditions imply the third condition: (a) P1S3,1 · P2S1,2 · P3S2,3 − S3,2P2 S1,2P3 S2,3P1 = 1; S1,2S1,1 · S2,3S2,2 · S3,1S3,3 (b) S1,1S1,3 S2,2S2,1 S3,3S3,2 =1;

(2) Prove further that (a) and (b) are equivalent if the Si,i are collinear.

Crux 2355.S25(5)318. (G.P.Henderson) For j =1, 2,...,m,letAj be non-collinear points with Aj = Aj+1. Translate every even-numbered point by an equal amount to get new points A2, A4, . . . , and consider the sequence Bj,whereB2i = A2i and B2i−1 = A2i−1.The laast member of the new sequence is either Am+1 or A m + 1 according as m is even or odd. Find a necessary and suﬃcient condition for the length of the path B1B2B3 ···Bm to be greater than the length of the path A1A2A3 ···Am for all such nonzero translations. Crux 1985 provides an example of such a conﬁguration. There, m =2n,theAi are the vertices of a regular 2n−gon, and A2n+1 = A1.

Crux 2356.S25(6)369. (V.Oxman) Five points, A, B, C, K, L, with whole number coordinates are given. The points A, B, C do not lie on a line. Prove that it is possible to ﬁnd two points, M and N, with whole number coordinates, such that M lies on the line KL and KLM is similar to ABC.

Crux 2358.S25(6)371*. (G.Leversha) In triangle ABC,letthemidpointsofBC, CA, AB be L, M, N respectively, and let the feet of the altitudes from A, B, C be D, E, F respectively. Let X be the intersection of LE and MD,letY be the intersection of MF and NE,andletZ be the intersection of ND and LF . Show that X, Y , Z are collinear. Solution: We use homogeneous barycentric coordinates with respect to triangle ABC,and interchange the labelling of the points X and Z. For convenience, write a := b2 + c2 − a2, b := c2 + a2 − b2,andc = a2 + b2 − c2.Since BD : DC = c cos B : b cos C = c2 + a2 − b2 : a2 + b2 − c2 = b : c, YIU : Problems in Elementary Geometry 378

D is the point with homogeneous coordinates 0 : c : b. Since the midpoint M of CA has homogeneous coordinates 1 : 0 : 1, the equation of the line DM is   xyz det  101 =0, 0 c b or cx + by − cz =0. By interchanging a, b,andx, y, we obtain the equation of LE:

ax + cy − cz =0. It is easy to ﬁnd the intersection of LE and MD as the point

Z = c(c − b):c(c − a):c2 − ab.

Similarly, Y = b(b − c):b2 − ca : b(b − a). and X = a2 − bc : a(a − c):a(a − b). Now, these points X, Y , Z are collinear since the determinant   a2 − bc a(a − c) a(a − b) det  b(b − c) b2 − ca b(b − a)  =0. c(c − b) c(c − a) c2 − ab

The equation of the line is

a(b + c)(b − c)2x + b(c + a)(c − a)2y + c(a + b)(a − b)2z =0.

In terms of a, b, c,thisis

a2(b2 − c2)2(b2 + c2 − a2)x + b2(c2 − a2)(c2 + a2 − b2)y + c2(a2 − b2)(a2 + b2 − c2)z =0.

Alternative solution: The points D, E, F , L, M, N are concyclic, all lying on the nine - point circle. The collinearity of X, Y , Z follows immediately from Pascal’s mystic hexagram theorem. See, for example, Pedoe, Geometry, Dover reprint, 1988, p.335.

Crux 2359.S25(6)372. (V.N.Murty) Let PQRS be a parallelogram. Let Z divide PQ internally in the ratio k : . The line through Z parallel to PS meets the diagonal SQ at X. The line ZR meets SQ at Y .FindtheratioXY : SQ. YIU : Problems in Elementary Geometry 379

Crux 2360.S25(6)374. (K.R.S.Sastry) In triangle ABC,letBE and CF be internal angle bisectors, and let BQ and CR be altitudes, where F and R lie on AB,andQ and E lie on AC. Assume that E, Q, F ,andR lie on a circle that is tangent to BC. Prove that triangle ABC is equilateral. Solution: The lengths of the various segments are bc bc AR = b cos A, AQ = c cos A, AF = ,AE= . a + b a + c · · c b Since E, Q, F ,andR lie on a circle, AE AQ = AF AR, i.e., a+c = a+b . From this, we have b = c. ab ac Now, CQ = a cos C,andCE = a+c = a+c . Since this circle is tangent to BC (necessarily at · a 2 a+c a its midpoint), CQ CE =(2 ) .Fromthis,cosC = 4c . Since cos C = 2c ,wehavea = c,and the triangle is equilateral.

Crux 2361.S25(6)374*. (K.R.S.Sastry) The lengths of the sides of triangle ABC are given by relatively prime natural numbers. Let F be the point of tangency of the incircle with side AB. Suppose that ABC =60◦ and AC = CF. Determine the lengths of the sides of triangle ABC. Solution: Let s =(a + b + c)/2. Note that BF = s − b.Since ABC =60◦,wehavefromthe triangles ABC and FBC,

b2 = a2 − ac + c2, (3) b2 = a2 − a(s − b)+(s − b)2. (4)

Upon subtraction, we obtain 1 0= (s − a)(a + b − 3c). 2 Since s − a =0,wemusthave b =3c − a.From(1),wehavec(5a − 8c) = 0. Since the sides are relative prime natural numbers, a =8,c =5,andb =7.

Crux 2365.S25(6)379. (V.Oxman) Triangle DAC is equilateral. B is on the line DC so that BAC =70◦. E is on the line AB so that ECA =55◦. K is the midpoint of ED. Without the use of a computer, calculator, or protractor, show that 60◦ > AKC > 57.5◦.

Crux 2366.S25(6)380*. (C.Shevlin) Triangle ABC has area p,wherep ∈ N.Let

Σ=min(AB2 + BC2 + CA2) YIU : Problems in Elementary Geometry 380 where the minimum is taken over all possible triangles ABC with area p,andwhereΣ∈ N. Findtheleastvalueofp such that Σ = p2.

Crux 2367.S25(6)381*. (K.R.S.Sastry) In triangle ABC, the cevians AD, BE intersect at P .Provethat [ABC] × [DPE]=[AP B] × [CDE]. Here, [ABC] denotes the area of triangle ABC etc.

Crux 2375.S25(7)436*. (T.Seimiya) Let D be a point on side AC of triangle ABC.Let E and F be points on the segments BD and BC, respectively, such that BAE = CAF.Let P and Q be points on BC and BD respectively, such that EP//DC and F Q//CD.Provethat BAP = CAQ.

Crux 2376.S25(7)437*. (A.White) Suppose that ABC is a right angled triangle with the right angle at C.LetD be a point on hypotenuse AB, and leet M be the midpoint of CD. Suppose that AMD = BMD.Provethat (1) AC2MC2 +4[ABC][BCD]=AC2MB2; (2) 4AC2MC2 − AC2BD2 =4[ACD]2 − 4[BCD]2.

Crux 2377.S25(7)438. (N.Dergiades) Let ABC be a triangle and P a point inside it. Let BC = a, CA = b, AB = c, PA = x, PB = y, PC = z, BPC = α, CPA = β and AP B = γ. − − − π Prove that ax = by = cz if and only if α A = β B = γ C = 3 .

Crux 2379. (D.J. Smeenk) Suppose that M1, M2, M3 are the midpoints of the altitudes from A to BC,fromB to CA and from C to AB in ABC. Suppose that T1, T2, T3 are the points where the excircles to ABC opposite A, B,andC touch BC, CA,andAB. Prove that M1T1, M2T2 and M3T3 are concurrent. Determine the point of concurrency. Solution: These lines are concurrent at the incenter I of the triangle, which has barycentric 1 coordinate 2s [aA + bB + cC]. Since BT1 : T1C =(s − c):(s − b), 1 T = [(s − b)B +(s − c)C]. 1 a Let P be the projection of A on BC.Since

BP : PC = c cos B : b cos C = c2 + a2 − b2 : a2 + b2 − c2, YIU : Problems in Elementary Geometry 381

1 2 2 − 2 2 2 − 2 P is the point 2a2 [(a + b c )B +(c + a b )C], and the midpoint of the altitude AP is 1 M = [2a2A +(a2 + b2 − c2)B +(c2 + a2 − b2)C] 1 4a2 1 = [2a2A +2abB +2acC − 4(s − a)(s − b)B − 4(s − a)(s − c)C] 4a2 1 s − a = (aA + bB + cC) − T 2a a 1 1 = [sI − (s − a)T ]. a 1 In other words, 1 I = [aM +(s − a)T ]. s 1 1

The line M1T1 therefore contains the incenter of the triangle; so do the lines M2T2 and M3T3.

Crux 2382.S25(7)440*. (M.Aassila) If ABC has inradius r and circumradius R,show that B − C 2r cos2 ≥ . 2 R Solution: We shall assume B ≤ C, so that the inequality is equivalent to A C − B √ 2R sin(B + )=2R cos ≥ 2 2Rr. 2 2 Suppose the bisector of angle A intersects the circumcircle at M.NotethatAM =2R sin(B+ A A r ). Also, IM = BM =2R sin ,andAI = A ,whereI is the incenter. Consequently, 2 2 sin 2 A √ √ 2R sin(B + )=AM = AI + IM ≥ 2 AI · IM =2 2Rr. 2 This completes the proof.

Remark. Equality holds if and only if I is the midpoint of AM. This is the case if and only if B C 1 tan 2 tan 2 = 3 .

Crux 2383.S25(7)441. (M.Aassila) Suppose that three circles, each of radius 1, pass through the same point in the plane. Let A be the set of points which lie inside at least two of the circles. What is the least area that A can have? See also Crux 2483. YIU : Problems in Elementary Geometry 382

Crux 2397. (T.Seimiya) Given a right - angled triangle ABC with BAC =90◦.LetI be the incenter, and let D and E be the intersections of BI and CI with AC and AB respectively. BI2+ID2 AB2 Prove that CI2+IE2 = AC2 . r r Solution: Let r be the inradius of the triangle. Then, BI = B and ID = B . It follows sin 2 cos 2 that 1 1 4r2 BI2 + ID2 = r2( + )= . 2 B 2 B 2 cos 2 sin 2 sin B 2 2 4r2 Similarly, CI + IE = sin2 C , and the result follows from the sine law.

Crux 2398. (T.Seimiya) Given a square ABCD with points E and F on sides BC and CD respectively, let P and Q be the feet of the perpendiculars from C to AE and AF respectively. CP CQ ◦ Suppose that AE + AF =1.Provethat EAF =45 . Solution: Suppose the square has unit side length. If CAP = θ<45◦,then CP √ 1 1 = 2sinθ cos(45◦ − θ)= + √ sin(45◦ − 2θ). AE 2 2 Similarly, if CAQ = φ<45◦,then CQ 1 1 = + √ sin(45◦ − 2φ). AF 2 2

CP CQ ◦ − ◦ − It follows that AE + AF = 1 if and only if sin(45 2θ)+sin(45 2φ) = 0. This is possible only when (45◦ − 2θ)+(45◦ − 2φ)=0,i.e., θ + φ =45◦.

Crux 2407. (C.J.Bradley) Triangle ABC is given with BAC =72◦. The perpendicular from B to CA meets the internal bisector of BCA at P . The perpendicular from C to AB meets the internal bisector of ABC at Q. If A, P , Q are collinear, determine ABC and BCA. Solution: These angles are 84◦ and 24◦. We begin by considering a generic triangle ABC, with sides a, b, c, and opposite angles α, β, γ respectively. The projection of B on CA has barycentric coordinates 1 E = ((a cos γ)A +(c cos α)C). b The bisector of angle C meets BE at 1 1 P = (E +(cosγ)B)= ((a cos γ)A +(b cos γ)B +(c cos α)C). 1+cosγ b(1 + cos γ) YIU : Problems in Elementary Geometry 383

Similarly, the bisector of angle B intersects the altitude from C at 1 Q = ((a cos β)A +(b cos α)B +(c cos β)C). c(1 + cos β) The points A, P , Q are collinear if and only if   100 det  a cos γbcos γccos α  =0. a cos βbcos αccos β

This reduces to cos β cos γ =cos2 α.Fromthis,

cos(β − γ)=2cosβ cos γ − cos(β + γ)=2cos2 α +cosα =1+cos2α +cosα. Now, with α =72◦,wehave 1 cos(β − γ) = 1 + cos 144◦ + cos 72◦ =1− sin 54◦ +sin18◦ = , 2 by Crux 2348. It follows that β − γ =60◦.Sinceβ + γ = 108◦,wehaveβ =84◦ and γ =24◦. This completes the proof.

Remarks. (1) From the above calculation, it follows that a unique triangle ABC exists with A, P , Q collinear for every acute angle α greater than 60◦. (2) Consider the counterparts of the line PQ for the other two pairs of vertices A, C,and A, B, we obtain three lines. These three lines are always concurrent, and the intersection is the point K on the line OI such that OI : IK = R : r. Here, O and I are respectively the circumcenter and incenter, and R, r the circumradius and inradius of the triangle. I omit the details, and perhaps shall propose it as a separate problem.

Crux 2415. (P.Yiu) Given a point Z on a line segment AB, ﬁnd a euclidean construction of a right - angled triangle ABC whose incircle touches hypotenuse AB at Z.

Crux 2416. (V.Konecny)

Crux 2417. (C.J.Bradley)

Crux 2418. (C.J.Bradley) In triangle ABC, the lengths of the sides BC, CA, AB are 1998, 2000, 2002 respectively. Prove that there exists exactly one point P (distinct from A and B) on the minor arc AB of the circumcircle of triangle ABC such that PA, PB, PC are all of integer length. YIU : Problems in Elementary Geometry 384

Crux 2422. (W.Janous) Let A, B, C be the angles of an arbitrary triangle. Prove or disprove that √ 1 1 1 9 3 + + ≥ 1 . A B C 2π(sin A sin B sin C) 3

Crux 2424. (K.R.S.Sastry) In triangle ABC, suppose that I is the incenter and BE is the bisector of angle ABC,withE on AC. Suppsoe that P is on AB and Q on AC such that PIQ is parallel to BC.ProvethatBE = PQ if and only if ABC =2 ACB.

Crux 2425. (K.R.S.Sastry) Suppose that D is the foot of the altitude from vertex A of an acute angled Heronian triangle. Suppose that the greatest common divisor of the side lengths is 1. Find the smalelst possible value of the side length BC,giventhatBD − DC =6.

Crux 2427. (T.Seimiya)

Crux 2428. (T.Seimiya)

Crux 2429. (D.J. Smeenk)

Crux 2430. (D.J. Smeenk)

Crux 2431. (J.Taylor)

Crux 2432. (K.R.S.Sastry)

Crux 2433. (K.R.S.Sastry)

Crux 2434. (K.R.S.Sastry)

Crux 2437. (Yiu) Let P be a point in the plane of triangle ABC. If the midpoints of the segments AP , BP, CP all lie on the nine-point circle of triangle ABC,mustP be the orthocenter of this triangle ? Solution: We use barycentric coordinates with respect to ABC. Denote the midpoints of AP , BP, CP by X, Y , Z respectively. The triangles XY Z and DEF are homothetic, the center of 3G+P homothety, being the midpoint of DX,isthepointK = 4 . The circumcenters of these two triangles are symmetric with respect to the center of homothety. If these triangles have the same circumcircle, then K is their common circumcenter. It follows that K is the nine-point center, YIU : Problems in Elementary Geometry 385

3G+H midway between the orthocenter and the circumcenter, or 4 , by the Euler line theorem. From this P = H, the orthocenter.

Crux 2438. (P.Hurthig) Show how to tile an equilateral triangle with congruent pentagons. Reﬂections are allowd. Compare Crux 1988.

Crux 2454. (G. Leversha) Three circles intersect each other orthogonally at pairs of points A and A. B and B,andC and C. Prove that the circumcircles of triangles ABC and ABC touch at A.

Crux 2455. (G. Leversha) Three equal circles, centered at A, B.andC, intersect at a common point P . The other intersection points are L (not on the circle center A), M (not on the circle center B), and N (not on the circle center C). Suppose that Q is the centroid of triangle LMN,thatR is the centroid of ABC,andthatS is the circumcenter of LMN. (a) Show that P , Q, R are collinear. (b) Establish how they are distributed on the line.

Crux 2456. (G. Leversha) Two circles intersect orthogonally at P . A third circle touches them at Q and R.LetX be any point on this third circle. Prove that the circumcircles of triangle XPQ and XPR intersect at 45◦.

Crux 2457. (G. Leversha) In quadrilateral ABCD,wehaveA + B =2α<180◦,and BC = AD. Construct isosceles triangles DCI, ACJ,andDBK,whereI, J, K are on the other side of CD from A, such that

ICD = IDC = JAC = JCA = KDB = KBD = α.

(a) Show that I, J, K are collinear. (b) Establish how they are distributed on the line.

Crux 2458. (N. Dergiades) Let ABCD be a quadrilateral inscribed in the circle centre O,radiusR,andletE be the point of intersection of the diagonals AC and BD.LetP be any point on the line segment OE and let K, L, M, N be the projections of P on AB, BC, CD, DA respectively. Prove that the lines KL, MN, AC are either parallel or concurrent.

Crux 2462. (V.N.Murty) If the angles A, B, C of triangle ABC satisfy

A B C cos A sin =sin sin , 2 2 2 YIU : Problems in Elementary Geometry 386 prove that triangle ABC is isosceles.

Crux 2464. (M.Lambrou) Given triangle ABC with circumcircle Γ, the circle ΓA touches AB and AC at D1 and D2, and touches |Gamma internally at L. Deﬁne E1, E2, M,andF1, F2, N in a corresponding way. Prove that (a) AL, BM, CN are concurrent; (b) D1D2, E1E2, F1F2 are concurrent, and that the point of concurrency is the incenter of triangle ABC.

Crux 2466. (V.Oxman) Given a circle (but not its center) and two of its arcs, AB and CD, and their midpoints M and N (which do not coincide and are not the end points of a diameter), prove that all the unmarked straightedge and compass construction that can be carried out in the plane of the circle can also be done with an unmarked straightedge alone.

Crux 2467. (W.Janous) GivnealinesegmentUV and two rays r and s, emanating from V such that angle(UV,r)= (r, s)=60◦ and two lines g, h on U such that (UV,g)= (g, h)=α, where 0 <α<60◦. The quadrilateral ABCD is determined by g, h, r, s.LetP be the point of intersection of AB and CD. Determine the locus of P as α variesfrom0to60◦.

Crux 2469. (P.Yiu)

Crux 2470. (P.Yiu)

Crux 2473. (M.A.Covas) Given a point S on the side AC of triangle ABC,constructa line through S which cuts lines BC and AB at P and Q respectively, such that PQ = PQ.

Crux 2477. (W.Janous) Given a nondegenerate triangle ABC with circumcircle Γ, let rA be the inradius of rhe region bounded by BA, AC and arc (CB) (s0 that the region includes the triangle). Similarly, deﬁne rB and rC .Asusual,r and R are the inradius and circumradius of triangle ABC.Provethat 64 3 ≤ ≤ 32 2 (a) 27 r rArBrC 27 Rr ; 16 2 ≤ ≤ 8 (b) 3 r rBrC + rC rA + rArB 3 Rr; ≤ ≤ 4 (c) 4r rA + rB + rc 3 (R + r).

Crux 2483. (V.Konecny) Suppose that 0 ≤ A, B, C,andA + B + C ≤ π. Show that

0 ≤ A − sin A − sin B − sin C +sin(A + B)+sin(A + C) ≤ π.

There are, of course, similar inequalities with the angles permuted cyclically. See Crux 2383. YIU : Problems in Elementary Geometry 387

Crux 2484. (T.Seimiya) Given a square ABCD, suppose that E is a point on AB produced beyond B,thatF is a point on AD [rodiced beyond D,andthatEF =2AB.LetP and Q be the intersections of EF with BC and CD respectively. Prove that (a) AP Q is acute angled; (b) PAQ ≥ 45◦.

Crux 2485. (T.Seimiya) ABCD is a convex quadrilateral with AB = BC = CD.LetP be the intersection of the diagonals AC and BD. Suppose that AP : BD = DP : AC.Prove that either BC//AD or AB ⊥ CD.

◦ ◦ ◦ 1 Crux 2486. (J.Howard) It is well known that cos 20 cos 40 cos 80 = 8 . Show that √ 3 sin 20◦ sin 40◦ sin 80◦ = . 8