The Radii by R, Ra, Rb,Rc

The Orthopole,

R. GOORMAGHTIGH,La Louviero, Belgium.

Introduction.

1. In this little work are presented the more important resear- ches on one of the newest chapters in the modern Geometry of the triangle, and whose origin is, the following theorem due to. Professor J. Neuberg: Let ƒ¿, ƒÀ, y be the projections of A, B, C on a straight line m; the perpendiculars from a on BC, ƒÀ on CA, ƒÁ on ABare concurrent(1). The point of concurrence. M is called the orthopole of m for the triangle ABC. The following general conventions will now be observed : the sides of the triangle of reference ABC will be denoted by a, b, c; the centre and radius of the circumcircle by 0 and R; the orthocentre by H; the centroid by G; the in- and ex-centresby I, Ia 1b, Ic and the radii by r, ra, rb,rc;the mid-points of BC, CA, AB by Am, Bm, Cm;the feet of the perpendiculars AH, BH, OH from A, B, C on BO, CA, AB by An,Bn, Cn; the contact points of the incircle with BC, CA, AB by D, E, F; the corresponding points for the circles Ia, Ib, Ic, by Da, Ea,Fa, Dc, Eb, F0, Dc, Ec, Fc; the area of ABC by 4; the centre of the nine-point circle by 0g; the Lemoine . point by K; the Brocard points by theGergonne and Nagel points by and N(2); further 2s=a+b+c. Two points whose joins to A, B, C are isogonal conjugate in the angles A, B, C are counterpoints. The images, through Am, Bm, Cm,of the points where a straight line m meets the 'sides, are on another straight line, which ' will be called reciprocal line to m, (1 Neubeitg, NCM,1875, p. 189; H. van Aubel, NCH,1875, p, 316; Des- boves Questionsdo Geometric,1875, p. 241;Neuberg, NOM1,1878,p. 379; Lemoino, ME, 1884,p. 50; Cwojdzihski, AMPh,3u-ser., voi 1, p. 174; Neuberg,J AMPh,sd ser.vol.3, p.89, Surles projections et eontreprojections d'untriangle, 1890, p. 74-76,M, 1911, p.244,M,1012, p.238; Gallatly, Themodern geometry of thetriangle, 2fla-ed., p. 46. (2) 78 R. GOORMAGHTIGH :

The straight line passing through the projections of any point Q of the circumeircle on the sides will be called Simson line of Q; Q is the pole of the Simson line. AmBm, Cm is the medial triangle of ABC; the parallels to BC, CA, AB drawn respectively through A, B, C form the antimedial triangle of A BC. The following abbreviations will further be used for mathematical journals referred to : AFAS, Association frangaiso pour l'avancement des Sciences AG, Annales do Mathematiques (Gergonne), AMPh Archiv der Mathematik and Physik, BB, Bulletin de l'Acadbmie de Belgique (Classe des Sciences), BME, Bulletin de Math6matiques 616mentaires, ET, Educational Times, IM, Interm6diaire des Math6maticiens, JME, Journal do Math6matiques 616mentaires (Bourget, de Longchamps), JV, Journal de Math6matiques 616mentaires (Vuibert), M, Mathesis, MB, Mtmoires in-8° de l'Acad6mie royal de Belgique, NA, Nouvelles Annales do Mathematiques, NCM, Nouvelle Correspondance Math6matique, QJ, Quarterly Journal, SL, M6moires de la SoclLte royale des sciences de Liege, TMJ, Tohoku Mathematical Journal, , WT, Wiskundig Tijdschrift.

I. General theorems on the orthopole.

2. The ortlaopole theorem. Draw m' through A parallel to. m and let ƒÀ•L, ƒÁ•L be the projections of B and C on m•L, p and q the

projections of ƒÀ•L on AC and of ƒÁ on AB; ƒÀ•LƒÏ and ƒÁ•Lq meet AAn, at Mb and Mc.Then MbpAnC, ƒÀ•LpƒÁ•LC, are cyclic.

Hence (Fig. 1)

Therefore . If ƒ¿, ƒÀ, ƒÁ are the projections , of A, B, G on m, the perpendiculars from a on BC,ƒÀon CA, ƒÁon AB are concurrent at M and AM is equipollent to Aoc. 3. .Distance from M to m. When m moves parallelto. itself, the figure ƒ¿ƒÀƒÁM remains -unchanged in shape andsize, H sliding along a straight line 8 perpendicular to m; the distance S from H to m remains unchanged. Let ƒÆa ƒÆb, ƒÆc be the angles which .the sidesmake with m, these angles being measured from m as axis, and in the same sense. Then THE ORTHOPOLE: 79

4. The ortlaopde M lies on the Simson line perpendicular to m. If. T be the point where Aa cuts the circumcircle and T•Lits projection on BC, then the Angle TAC is equal to 90•Ž-ƒÆb, and the angles BCT and BAT are equal to 90•‹-ƒÆc.

Since TC

=2Rsin TAC=2RcosƒÆb,

TT=TOsin BCT

=2Rcos ƒÆbcos ƒÆc=AMb

Hence MT•L is parallel to AT, and, by a known property, T•L is the Simsonline M perpendicular to m. 5. Other Simson lines passing througla M. Let Q. and Q•L be the points where m meets the circumcircle; the perpendicular QQ, from Q on BC meets again the circumcircle at Q2. If Q" is the projection of T on QQ2, Qƒ¿TQ" is cyclic; hence the anglesAQ2Q,ƒ¿TQ, ƒ¿Q"Q are equal and AQ2 is parallel to ƒ¿Q" and also to MQ1, since ƒ¿M=

TT=Q"Q1. Therefore MQ1 is the Simson line of Q.

When m meets the circumcircle at Q and Q', the-Simson lines of Q and Q' pass through M(2); when m is tangent at Q to the circumcircle, the orthopole of m is the point where the Sum son line of Q' touches its envelope (S to z ne r s tricuep).

6. Orthopole and pedal triangles. Let XYZ and X'Y'Z' ,be any two triangles; then a certain point V, called of the two triangles has the same relative barydentric coordinates in ., XYZ and X 'Y'Z' . if. X, p, v are these coordinates, V isthe. centre of gravity

(1) Neuberg, Sur lea cercles podaires, BB, 1910; de Lepiney, M, 1914, p. 177. (2) Neuberg, Sur lea projections et contre-profa Lions d'un triangle, MB, 1890; goons M, 1896, p. 57. Hence, 11 is the orthopole of 130, CA, AB. (8) Neuberg, Sur lea equicentres de deux systentes de n points, SL, 1918. R. GOORMAGHTIGH:

of masses ƒÁ, ƒÊ, v at X, Y, Z and at X', Y', Z', Therefore, V

being any point,

Hence, the vectors being in the same plane,

Let now PaPbPc and SaSbSc be the pedal triangles(1) of two points P and S on m, in the triangle ABC and .denote by M equicentre of PaPbPc, SaSbSc; then

We will provo that this new point M is the orthopole of m. If

3 is the pedal triangle of a, ƒ¿1ƒ¿2ƒ¿

3:ƒ¿ƒ¿1ƒ¿3= ƒ¿ƒ¿3sin B :ƒ¿ƒ¿2Sin C=ƒ¿ƒ¿1ƒ¿ b•Aƒ¿cosƒÆc : c•AƒÆcosƒÆb =b secƒÆb : c secƒÆc:

Hence .M lies on ƒ¿ƒ¿1; and so for the perpendiculars from ƒÀ on CA

and from ƒÁ on AB. The orthopole M of m has the same relative barycentric coordinates

a see ƒÆa, bseeƒÆb, cseeƒÆc in the pedal triangle of any point P on m (2).

In section 27, we will find that, for any straight line m, .

Hence the absolute harycentric coordinates(3) of the orthopoteM in the pedal triangle. of any point -on m are (4 )

7. Orthopoleand pedal circles. Lemoyne's Theorem. Analytical proof(5), Let Pa, Pb, Pc be the lengths of thesides of the pedal triangle PaPbPc of a point P on m; the barycentric equation to the circle PaPbPc in the "triangle PaPbPc is (1) Trianglesformed by the projectionsPa, Pb,Pcand Ba,Bb, Be of Pand.s on BC,CA, AB respectively. (2) Neuberg, Fiurleg cercies poZalres, 1313, 1910. (3) For absolutebarycontric coordinates, , 4) Goor'masrhtigh. M. 1925.p.197. ( (5) Lemoyne's theoremhas beenpublished without proof in riA, 1904,p: 400. Proof3were given by N e u b e r g (anal.,)138, 1910, Lienard (anal.),M, 1912,p. 288, 1014p. 38,dQ Lepiney (geom.),M, 1914, p. ,177,Gallatly (geom.),Modern geometry of thetriangle, Sd ed, p. 51, Bouvaist (geom.),NA, 1915, p.'555, Thebault (geom.), NA, 1918,p. 293,TMJ, Vol19, Goormaght1gh (anal.),M. 1925,p. 196(the here givenproof).

80 THE ORTHOPOLE. 81

and it is known that the power, for the circle PaPbPc, of 'a point M, whose absolute barycentric coordinates in PAPBPC are ƒÁ, ƒÊ, v, is

But Pa,Pb,Pc being the tripolar coordinates PA,PB,PC of P,

d being the distance from m to 0. Hence

The orthopole J of m has the same power witli regard to the pedal

circle of any point P on m; this power is twicethe product of the distances from m to 0 and to N(2). d and 8 will be taken in the same sense when 0 and . Mare on

the same side of m.

8. Geometrical proof of Lemoyne' s theorem (3 ). Let J be one of

the points where the pedal circle of P meets m;the perpendicular g

at J to m is a tangent to the conic 1 inscribedin ABC, having a

focus at P, the pedal circle being the auxiliary circle of X. Hence,

if g cuts the sides of ABC in a1, b1, c1,-'the mid-points a1, b1', c1' of a1, Bbl, Cc1, are on a straight line going through. the centre of A2

(Newton); this centre is also the centre of thepedal circle of P

(Fig. 2). . . When g moves, parallel to itself) a1 bl' c1' remains - tangent to a

parabola _(7r), inscribed in the medial triangle, the directrix being the circumdiameter t parallel to m; in section 27, we will prove "that the

focus of is the orthopole of t. If the distanced from J W . t is denoted by JJ", JJ is the. image of w through al•L b1•Lc1•L and. J•Lw is

parallel to (JM, since wM= J•LJ But JM meets the pedal'circle of P again at J" the image of

(1) Gallatly, Modern geometry of the triangle,2d-ed., p. 11. 2) This theorem is a special case of the follwing ' theorem( due to Bricard, NA, 1920, p: 200: . f S1,S2, S3, are the tangential equationsI to three conics, the auxiliary circles of the conics k1S1+k2S2+k3S3=0, hautng a focus ion a given straight line, are orthogonr,3 to a fixed circle; A proof has been given by M.-F. Egan, NA, 1923-1924,'p.- 354. (8) Suggested by a proof given by R,. Bouvaist,N A, 1915, p. 555. 82 R. GOORMAGHTIGH:

J through a1'b1'c1'and w, J'', J, J' are on a circle meeting again Ma at f, the image of to through. the projectionofthe mid-point of JJ' on MW. Hence Mf=2ƒÂ; therefore the power of ' M for the pedal circle of P is MJ.MJ"= Mf•112as=2d8. Another form of Lemoyne's theorem may here, bo mentioned : The orthopole of m hasM same

power with .regard to the ,pedal circle of any point on the counter point conic of m. When the counter-point conic is a rectangular hyperbola (i. e. when m is a circumdiameter), all the pedal circles pass through the same point, centre of the hyperbola (1). For instance, the Nage1 point is on Feuerbach'shyperbola., counter-point conic of 10; hence the pedal circle of the Nagel point pass through the Feuorbach point (orthopole of 10)(2). 9. Other general theorms. The preceding proof shows that M lies on the perpendicular from J on air bigcit. If J is any point on m, the perpendicular frontJon the reciprocal line of the perpendicular on m at J passes through, the oAhopole M. Let now Va, Vb, Vc be the points where m cuts .BC, CA, AB; then the common radical axis of the circles having AVa, BVb, CVc, as diameters passes through M, and also, as known from the geometry of the quadrilateral, through the orthocentres of the four triangles of the quadrilateral formed by ABC and m. In a quadrilaieral, the orthopoles of eiichaofthe side for the triangle formed by the three others are four points on astraight line passing through (1) Bobillier,, AG, 1829,p. 349; see also section29. ' (2) Meytelon, JV, 1913-1914,p. 92. THE ORTHOPOLE. 83 the orthocentresof thefour trianglesof the quadrilateral(1). But the orthocentre-line is perpendicular to the reciprocal line of m in A.BO: The orthopoleR of a straight line m in a triangle ABC lies on the perpendicularfrom the orthocentreof ABC on the reciprocal line of m. 10. Orthopolesof a straight linein thefour trianglesof a quadrilateral. The result just obtained may be generalized as follows: The orthopolesof a given straight line m in all the trianglesformed by n tangentsto a parabola are on a straight line. Let ILDbe the focus of the given parabola, l its vertex-tangent. The circumcircle to any of the triangles' formed by three of the n tangents passes through c, and 1 is the commonSimson line 'of 4) in all the triangles. Draw through (P a parallel m1 to m; since the orthopole of m1 in any of the triangles lies onthe Siinson line of in this triangle, the orthopoles of m1 in all the triangles lie on 1. But when m moves parallel' to itself, the orthopole slides along a perpendicular to m, with the same speed as m. Hence, the orthopolesof m in the trianglesformed by n tangentsto a parabola are on a straight line 1,parallel to the vertex-tangent,1, the distancefrom Zto Z beingequal to theprojection, on the axis, off the distance from thefocus 0 to the given straight line m. Any four straight lines being tangent to a parabola, we find the following theorem : The orthopoles of a straight line m - in the four triangles of a quadrilateral are on a straight line (2 ).

Considering the spool al case when m is a side of quadrilateral , we find that the orthopole of each side in the triangle formed by the three others is on the straight line passing through the orthocentres of the three triangles to which this side belongs ,i.e to the orthocentre-li ne.

11. Other constructions of the orthopole. Returning to the general case concerning the equicentre of two trianglesXYZ , X'Y'Z' (section 6) , let. be the points (YZ, Y'Z'), (ZX, Z'X'),(XY, X' Y'); if are the points where the sides of the antimedial triangle of XYZ meet those of the antimedial triangle ofX•LY'Z' , the straight arcs concurent, the. point of concurrencebeing lines the

(1) Goormaghtigh, NA, 1919, p. 39; Servais, M, 1923, p. 152; Opper- nan , Le qucu2rilaOre oom pt et , 2d-ed., p. 69;do Lepiney ; M, 1922, p. 314. (2) For this special case, see Servais , M, 1923, p. 12. 84 R. GOORMAGHTIGH:

equicentre V of XYZ and X•LY'Z'. Denote by ƒÌ1, and ƒÌ2 the points (XV,YZ),(X'V; Y'Z'); then

and V has the same barycentric coordinates in XYZ and X'Y'Z'. If thecorresponding sides of thepedal triangles PaPbPc, SaSbSc, of any two points on m meet at and those of the antimedial triangles of PaPbPc,SaSbSc, at then are concurrent, the paint of concurrence being the orthopole .M of m (1). Let further Wa be the point (m, AO)2 then the join of the pro-

jections of wa on AB and AC is parallel to BC,and the preceding theorem gives the following property : If wa, wb, wc, are the points where m meets AO,B0, C0, and PaPbPc the pedal triangle of any point P on m, the orthopole M lies on the straight lines passing respectively tlaro24gh Pa, Pb, Pc,and through the points where PbPc,PcPa, PaPb meet the joins of the projections of cua on AB and AC, of wb on BC and BA, of wc, on CA and CB (2). 12. The reciprocal line of the image m2 of m through the circumcentre passes through M (3 ). Denote by 0 the Siinson line perpendicular to m, containing the orthopoles of all parallels to m and meeting the sides at ƒÆa, ƒÆb, ƒÆc,. It is known that if a straight line moves parallel to m, the reciprocal line envelops a parabola inscribed in ABC and having ƒÆ as vertex- tangent. Let q, and q2 be the points where the moving line and its reciprocal line meet ƒÆ; then Now Aq,Bq,Cq being the proection s of A, B, C on ƒÆ, when

q1•ßAq, then q2•ßƒÆa, and so for Bq and Cq; butthe mid-points of AƒÆa, BƒÆb, CƒÆc,are on a straight line perpendicular to ƒÆ (locus of the centres of the conics inscribed in ABC and tangent to 0). Hence AqƒÆa, BqƒÆb, CqƒÆc, have the same mid-point w.Therefore (q1,..) and

(q2, ....) are symmetrical, the centre being w.In section. 34, we will prove that the reciprocal line of a circumdiameter t passes through the orthopole w of t. When the parallel to m is7112, then q2;w is equal and opposite to the distance from m2 to tand therefore equal to the distance from m to t; hence q2 is then the orthopole 31

of m. 13. The orthopoleM of m lies on the incircle or on awe of the e.ccircle

(1) Neuberg, BB, 1910. (2) Neuberg, Ibfd, (3) Goormaghtigh, M, 1914, p. 160. THE ORTHOPOLE. 85, of the triangle formed by the images of m through the sides of the medial triangle(1). Let ua, ub, uc be the points where m cuts BmCm , CmAm,AmBm the images of m through.BmCm, CmAm AmBm,form atrangle aoboco. Then AnBm and BmCm being bisectrices of boucuaand bouauo Bmbo is a bisectrix of uabouc; hence Amao,Bmbo, Cmcomeet at a point co, centre of the incircle of aoboco(supposing all anglesin ABC acute) (Fig . M... Now the image Oaof 0 through Bn: Cm is on the nine-point circle, and the distance from 0 to m is equal to the distance from Oa to boc0; hence the parallels from Oa, 0,, O , images of 0 through B. C., C.A., AmBm, to boco) coao, aobo, pass through w. Further, by a known property, these parallels pass through the point of the circle ArBm Cm, whose S i m- son line in AmBmCm is parallel to m; hence tB is that point , and we will find, in section 27, that the pole of the A„tiBmCm S i m s o n line parallel to m is the orthopole in ABC of the circumdiameter t parallel to m: Finally the distance between the orthopoles of m and t is equal to the distance from m to 0 , i.e. the radius"ofthe incircle of aob0cp, which proves the theorem . If A is obtuse, then m' will be the ao ex-centre in aoboco. 14. Another general theorem may also be mentioned here : When two triangles are inscribed in a circle and circumscribed to a parabola, tine mid-point. of the distance of their orthocentres is the orthopole of any side of one -of the triangles for the other( 2) . (1) This theorem does not seem to have been mentioned d yet . ) C oJuzinskY, Aiririti, 1902, P . 316; de LePney, .M, 1914,(2 p. 179.

i 86 R. GOORMAGHTIGH:

It is known that in a system of poristic•. triangles,' inscribed in a circle and circumscribed to a parabola, the S iinson-lines for a given point on the circle are all parallel(1). Hence,if ABC and . A1B1C1 are two of the triangles, the Simson line of B in the triangle A1B1C1 is parallel to the perpendicular BBn, from B onCA and passes through the mid-point of Bh1, or HH1, H and Hi being the orthocentres of ABC and A1B101; the Simson line of C in the triangle A1B1C1

passes also through the mid-point of HA. Hence the orthopole of BC in A1B1C1, is the mid-point of HA. 15. Normal coordinates of the ortliopole. Let lx+my+nz=0 be the ABC normal equation to m and denote Ace, BƒÀ, CƒÁ by da, db, dc; then and

Further

The normal ABC coordinates of the straight linewhose normal equation inABC, is

are as

II. Some loci and envelopes connected with the orthopole-geometry, 16. Orthopoles of straight-lines through a fixed ,point. We have seen, in section 6, that the •barycentric coordinates X, 'pt, v' of .Afin (1) Servais, M, 1922,p.268. (2) Ga11athy, Themodentrn geometry of the triangle,2d-ed., p. 48. THE ORTHOPOLE. 87

the pedal triangle PaPbPc of any point P on m are as a sec ƒÆa, b sec 8b,

c sec eo. But it is known that, for any straight line,

hence (1)

When m goes through a fixed point P, the locus . P(P) of the orthopole of m is a conic passing through the pro M , jeetions of P on the sides. When m is parallel to BC,. the orthopole of m is on AH and

at a distance from H equal to P,P, H being the orthopole of 13C. The centre of the conic (P) is the mid point ofthe distance from P

to the ortliocentre H(2). We will now prove that the orthopoles cor-

responding to two perpendicular lines m and ml are on the same diameter

of (P).

In section 27, we will find that the orthopolesof two perpen-

dicular circumdiametors - are on a same diameter of the nine-point

circle. Let then t and t, be the eircumdiameters parallel to m and m,, w and m', their orthopoles; as the distances from t to 112' and from tv, to M, are equal to those from m to t and from mi to 4 , and as 0,, is the mid-point of HO and of mm,, the mid-point of HP is also the mid-point of 2 M (3 ).

17. Properties of a group of conies. The geometry of the conics

(P) is very interesting,. a ,pd a few theoremsmay be briefly mentioned here.

The barycentric equation of (P) in the trianglePaPbPc being only expressed in terms of a, b, c,, the same occurs for the barycen- trio coordinator of -its centre in PaPbPc. If P is any point in the plane of ABC, then themid paint of the distance from P to the orthocentre of ABC has always the same barycentric coordinates sin 2A, sin 213, sin 2C in the pedal triangle of P.

The barycentric equation of (P) shows further(4) that , if the pedal triangle PaPbPc of P is projected on a plane into a triangle similar to ABC, the conic r(P) is projected . into a circle.

Hence 2p, 2q being the axes of . 1(P), and ‡™' the area of PaPbPc(5);

(3) This equation has been given, with another proof, by JO 'N 8 ,u b e rg, BE, 1910. (2 ) Goormaghtigh, M, 1914, p. 269. (3) The ban It, 111, 1923, p. 144; ervais, M, 1923, p. 149.. (4) Gal1at1y, Modern geometry of the triang'e, 2rd-ed.,p.108. (5) Gallatly, Ibid., p. 108. 88 R: GOORMAGHTIGH

The sum of the axes of any of the tonics (P) isequal to the circumdiameter of ABC. Further, the power of P for the circumcircle ABC is

Hence (1)

The product of the axes of the conic (P) is equal to the power of P for the circumcircle ABC.

Hence, the difference of the axes of (P) is equal to 2 OP.

18, Ort1iopolar triangle of a point. Through any point M pass three Simson - lines; hence (section 5) there are three straight lines, called orthopolar lines of 1, having M as orthopole, and they form a triangle -inscribed in the circumcircle ABC, since the vertices of this triangle are the poles of the three Simsonlines. This triangle is called the orthopolar triangle of M in the triangle ABC. We will -now find the envelope of the orthopolar lines of 31, when m moves along a straight line ll. Let t bethe Simsonline

parallel to 11 and (D its pole; it is • known (sections 4, 5) that the orthopolar lines of points on l are those perpendicular to 1 and those going through (D. To find the orthopolar lines of points on 4, we proceed as follows : any straight line through ƒÓcuts l in v, the

perpendicular on by at V cuts 4 at vi, the perpendicular vlv2 on vvl at Vl will then have its orthopole on h since (Dv has its orthopole on

1, and the distance of these two orthopoles is equal to vv,. Denote

by v. the projection of 4) on V1 V2; the projection of v1v2 on a perpendicular on 1 is equal to that of v(D and it is known that the mid- point of ll is on 1. Hence We find the following theorem : (1)I

t may be mentioned here -that any conic touches Steiners tricusp in three points (Goormaghtigh, NA, 1014, p. 121). When P is the image of If through G, P(P) is the locus of the centre of gravity of the projections of a moving point of the circumeircleaon the sides of the triangle (Goormaghtigh, NA, 1915, p. 452. THE ORTHOPOLE. 89

The orthopdar lines of a ,point M, moving on a given straight line l1, envelop a parabola ‡U1. The directrix is theparallel to l1,,' through the image of H through l1; the focus is the pole ƒÓof the Simson line l parallel to j (1): The orthopolar triangles ofthe points on a straight line li form a system of triangles . circumscribed to the parabola IT and inscribed in the circtuncircle ABC. When 4 is a Simson line, the orthopolar lines are those perpendicular -to 1, and those passing through the pole of the Simson line. ƒÓ 19. Two particular cases. When li passes through the orthocentre H,‡U1 is inscribed in ABC. If l1, is the Eu1er-line OH, ‡U1 is Xiepert's parabola ( 2 ); but it is known that this parabola is :tangent to the Lemoine axis (3 ). The orthopole of the Leinoine axis is on the Eul er lines (4). . When l1=HK ‡U1, is the inscribed parabola having HK as directrix; this parabola is tangent to the radical axis' of the circum- and nine-point circles. The orthopole of the radical axis of the circum-and nine point circles lies on the straight line joining the orthocentre to the Lemoine paint. 20. Locus of the ortlurpoles of the Simson lines. When m is a Simson line, the reciprocal line of m is the Simson line perpendicular on m; hence, the final theorem in section 9 shows that the orthopole of m is the projection of H on the reciprocal line of m. If ƒÓƒÓ•L is a diameter of the circumcircle, the orthopole of the Simson line of ƒÓ is the projection of the orthocentreH on the Simsom line of ƒÓ(5). The locus of the orthopoles of the Simson linesof a traingle is the pedal curve of Steiner's tricusp for the orthocentrr H. It follows also that the distance from the orthopole M of m to m is equal to the distance from the orthocentre H tothe Simson line parallel to m(6)

(1) Neuberg, BB, 1910. (2) Inscribed parabola having Off as directrix. (3) Polar of the Lemoine point for the circumcircle. (4) Goormaghtigh, M, 1914, p. 151. (s) Goormaghtigh, M, 1.914, p. 162, ET 1915, p.39, Youngman, Davis, TE, 1915, p. 164, Servais, M, 1923, p. 149. (6) If two triangles have the same ,steieiner;stricusp and if through their orthocentres two parallel straight lines are drawn, the orthopole of the first line for the second triangle is the same as the orthopoleof the second line for the first triangle. To prove this theorem, consider the tangent to the tricusp parallel tothe two lines, use preceding property and note that the orthopoles are on the tangentto the tricusp perpendicular to the given lines (Servais, M, 1923, p. 150). 90 R.GOORMAGHTIGH:

21. Locus of the orthopoles of the tangents to the circumcircle.

Referring to the final theorem in section S, wefind the following

result:

The locus of the orthopoles of the tangents to the circumcircle is Steiner's

trieusp (1).

22. Locus of the , orthopoles of a given line in a system of poristic triangles. When a system of poristie triangles'are inscribed in a circle

0 and described about a conic, the Simson linesof given point on

the circle for all the triangles pass through afixed point(2); it is

further known that the angle of the Simson lines for two points

and, ƒÓ' ' is ,

The locus of the orthopoles of a given straightline in a system of

triangles inscribed in a circle and decribed about a conic is a circle(3).

When the given conic is a parabola, the Simson lines of a given

point on the circle are all parallel; hence thelocus of the orthopoles is then a straight line(4).

23. Locus of the image of M through m when m passes through a fixed point. Let M' be the image of M through m; m passes through a fixed point P, whose pedal triangle is PaPbPc. Let further be cc,

R, ƒÁ the projections of A, B, C on m, h1, h2,h3 the points where m meets AH, BH, CH, h1', h2', h3' the mid-points of Ah1 Bh2, Ch3; then

M is the point of concurrence of ah1', Bh2', ƒÁh3'. But the envelopes

of are the circles having their centres at the mid-points P1, P2, P3 of PA, PB, PC and tangent respectively to AH, BH, OH. The angles of and and, and

are equal to A, B, C. Hence ' the locus of M' is a limacon, being the isoptic curve of 112 and Ps for the angle A, of f and I , for the angle B, of and for the angle C (Fig. 4). When m passes through a fixed point the locus of the image of the orthopole m is a limacon(5).

This limacon is a conchoid of the circle P1P2P3; if P2M", P3M" are parallel to and the constant modulus of the conchoid is but if is the mid-point of PH, the distancefrom

(1) Generalisation: When a straight line m remains tangent to a circle concentric to the circumcircle, the locus of the orthbnnh of m is a frochoirial curve (Polid NA, 1920, p. 160). (2) Bickart, JV, 1910-1911, p. 165. (3) Goormaghtigh, M, 1914, p. 161. (4) Servais, M, 1922, p. 268. (5) Goormaghtigh, NA, 1918, p. 242. THE ORTHOPOLE. 91 to is the radius of the circle A and therefore equal. to the projection of P'Hon AB; the same applies to thedistance from P2M" nd the nroiection of P'H a on AC, Hence M'M"=PP'. The double point a of the limacon is the point where MIN" meets the circle P1P2P3, are as the distances from to and from to hence as the projections of P'H on A C and AB. Therefore OF,, are as cosƒÆƒ¿, cosƒÆL, cos ƒÆc, ƒÆa, ƒÆb,ƒÆcbeing the angles of PH with the sides of ABC. Let now t be the . circumdiamne- ter parallel to PH; we will find, in seetion 27, that the orthopole w of

t is on the nine-point circle AmBmCm, and that the distances from w to. Am, Bm.Cm are as cos ƒÆa, cosƒÆb, cosƒÆc. But P1, P2, P3 are the images of Am, Bm, Cm through the mid-point of OP' ; hence is the image of v through the mid-point of OP'. The limacon corresponding to point P is a conchoid of a circle equal to the nine point circle, having its centre at the mid point of OP, the constant modulus being half the distance from Pto the orthocentre H. The double point is the izaage of the ortizopvle ofthe cir cundiamc er. parallel to PH, through the mid point of the distance from 0 to the mid point of PH. When m passes through the orthocentre, the locus of the image of M through m is the nine-point circle (1).

Ill: Orthopoles of the eircumdiameters.

24. Denote by m' the orthopole of a cireurndiameter t, and by

n, ƒÆb, ƒÆe the angles of t with BC, CA, AB; then the preceding ƒÆ results lead to the following theorems: The orthopole w of a circumdiameter t is on thenine point circle(2) (section 7);the pedal circleof any point on t is passos through w (section)7;

the normal coordinates of vi in AmBmCmare as see ƒÆa, secƒÆb, seeƒÆe (sec-

(1) See section 8l. (2) reuberg, .rof ection et centre-projectionsd'un triangle f xe, MB, 1890, p. 76; Soons. M, 1896, p. 57, Cwoidzinski, AMPh, 34 ser., vol 2, p. 178. 92 R. GOORMAGHTIGH: tion 6), and therefore die distances tuAm; tvB,, to Cm are as, Cos ƒÆa, Cos db, cos 8a ; the reciprocal tins of t passes tlarougia w and T's perpendicular to tuff (sections 9, 12); if PaPbPc is the pedal triangle of any point Pont , the loins of Pato (PbPc,BmCm) Pb, to(PcPa CmAm), P, to (PaPb , AmBm) intersect at w (section 11). All these theorems and many others will now be obtained by direct methods. 25. Pedal circle Let Q and Q' be the points where t meets the circumcircle; the Simson lines of Q and Q' are rectangular and meet at a point w. Denote by Q1 and Q1' the projection of Q and Q' on BC, and by PRP,PC the pedal triangle of apoint P on t. If Q2 is the point where QQ, meets again the circumcircle, tta'Q1 is parallel ' to .AQ.; the angles AQ 0, AQZQ', wQ1Am are equal, and Am being the mid-point of Q1Q1', the isosceles triangles A OQ and tA,,Q1 are similar. Hence (Fig. 5) the triangles FAQ and PawQ1 are similar and theangles PAQ, PawQ are equal ; similarly PBQ=F,rJ7Q1. But

Hence thereforethe circle PaPbPcpasses throughw. When PM•ß0, PaPbPc is the nine-

point circle; Hence, this circle is the locus of w. 26. Generaliza- tion of Peuerbacla's theorem. Any two

counter-points P and P' being the foci of an inscribed conic, have the same pedal circle

(auxiliary circle); this circle cuts the nine- point circle in two

points, the orthopoles of OP and OP'.

(1) ollert1nsky, M, •1891,'p. 11 Fonten6, NA, 1906, p.55; Alizi, JV,1909- 1910, p. 126. THE ORTHOPOLE. 93 Hence,if the pedal circle is. tangentto the nine-pointcircle, 0, P and F are on a straight line. Whena point P, its counterpoint .P' and the errettmcentre0 are on a straightline, the pedal of P and P is tangentto the nine-point circle, thepoint of contacthe'ng the orthopoleof POP'(1). 27 . In the triangle Q1wQ1', hence AAnw=0 . Further if H1' is the tnid-pointof AH,Hl'Amw=6a and this angle is therefore equal to the angle of OAm and the perpendicular from A, on t. Hence 'W is the focus of a parabola inscribed in AmBmCm, having its axis perpendicular to t. Since 0 is the orthocentre of AmBmCm, t is the directrix of . Then the image a of a through BmCm, lies on t; but AnAa being equal to WAhA or ƒÆa, a is the projection of A on ' t. The distances of the orthopole w of t to Am, Bm, Cm are B cos ƒÆa, Boos ƒÆb, R cosƒÆc; AAh is the bisectrix of theangle between Ahw and the perpendicular from Ah, on t (3); w is the focusof a parabola inscribed in the medial ,triangle having t as directrix the Simson line of w in AmllmCt is parallel to t; the images of nr through BmCm, CmAm, AmBm are the projections a, ƒÀ, ƒÁ of A, B, C on t(l) ; the distances Aa, BR, •Cy from A, B, C to t are respectively equal to zv.A3, WB3, wCh. It follows also that, if t and It are two circumdiaineters and w1 their orthopoles, the angle wAhw1 is equal to the angle between t and 11. Hence wOvw1 is twice the angle between t and t1. If t and t1 are per-

pendicular, their orthopoles are on a same diameter of the nine point circle. The perpendicular from Am on t is isogonal to Amw in BmAmCm: Hence, in the triangles ABhCh, BChAh, CAhBh,similar to ABC, the straight lines, homologousto t, intersect act to (5). (1) Weill,. NA, 1880,p. 250; Mac Cay, Irish Academy,1889; Font n 6, NA, 1906,p. 606; Vacopuant, JV, 1910-1911,p. 161.See also section77. (2) SinceW Am, WBin, Worn are respectivelyequal to R cosOa, R cos03, R cosOc, by expressingthat the sum of the areas Bmtw'C,n,CmwAm, AmWBrn Is Jd, we find that for any straightline m,

(3) Generalization of a theorem due to Mannheim, for the F o ue r ba c h point r,JV_ 1909491&100 43) ; A,tm contains the mid-point of the - distance of the points whereA. perpendicular to i meets AAr,,and BC(Visschere, M.1914,p. 278; Thebault, M, 1922,'p. 291). (4) For the special case of the Feuerba h point; see: Briard, NA, 1906, p. 96 Bouvaist, NA, 1906,p. 600. (5) Goormaghtigh, M, 1922,. p. 403 (generalization.of a theorem due to Visschers, M, 1922,p. 287). 94 R. GOORMAGHTIGH: Call again Va the point where t meets BC, P anypoint on t, Pa its projectionon BC, p' the projection of Va on AP; thenp'F,, passesthrough w. For the angle offt and BC isequal to Pp'Pa arid to AA,w, and A, Vd,p', Ah, w are on a circle(1). 28. To determinethe points of contactof (ir) with the side of AmBmCm,we note that if Ah' is the .image of Ah through 07, A,,' is also the image of 0 through B. Cm; Ah'wmeets then B. C. on t and is perpendicularto Ahc'. Thecontact points of theparabola with the sides of AmBmCm lie on wAh,wBh, w Ch (2). These points are also on the perpendicularsdrawn from A, B, C on t. 29. The orthopolew as centerof the counter-pointconic of t. There is a conic passing through A, B, C and havingas asymptotea given Simson line. By a known property,the other asymptoteis the reciprocalline of the first. Hence, if Q is the poleof the given Simson line, the secondasymptote will be the Simson line of the point Q, whereQO meets again the circumcircle;hence the conic is a rectangular hyperbolaand its centre is w, the orthopole of QO. But if Qs is the point where the perpendicularfrom Q on B C meets again the circumcircle,the angles Q2ABand CAQ'are equal; hence, the counter-pointsof Q' and Q are the points at 00 on the Simson lines of Q and Q. Therefore,the rectangular,hyperbola is the counter-pointconic of QOQ'. The counter-point.conic of t is a rectangularhyperbola (H) havingw as centreand as asymptotesthe Simson linesofthe points wheret meets the circumcircle(3). It followsthat, if P' is the counterpoint of anypoint on t, thenine-_ point circlesof .BP'C, CP'A, AP'B pass throughw. Further P' being on (H), in the triangle formedby (AP', BC), (BP') CA), (CPI, AB) each vertex has the oppositeside as polar for (H). If P' is the ' counterpaint of any point P on t,' the circlepassing

(1) Alisi, IV, 1910-1911,p. 176, IV, 1911-1912,p. 7. (2) For the special case of the Fenrba h point,see: Thebault, IV, 1912- 1918, p. 2: M, 1913, p. 180:. for the general case, Goormaghtig li, M. 191$. n. 1951_ (3) MoyteIon, IV, 1911-1912,p. 7. Bobi11ier had already proved (AG, 1829, p. 349) that the pedal circles of the 'points of a rectangular hyperbola with respect to an inscribed triangle pass through the centre. THE ORTHOPOLE, 95 through the points (AP',BO), (BP', CIA),(CPI,AB) passes also through the orthopole a of t (1). For instance, the Lemoine point of IaIbIc, has as ,counter-point in ABC the centre of similitude of DEF and h4b1Q,and this point is on 1C. Hence, the Feuerbach point (orthopoleof IU) is on the circle passing through, the points where BC, CA,.AB meet the lines ,joining A, B, C to the Lemoine point of 4,f,,(1). Similarly, the circles passing respectivelythrough, Da, Efb,F,, and (Al, BO), (Br, CA), (CI, AB) pass also through the Feue rba c h point ($ ). On the cireumndiameter t, there arr two points,inverse in the circle ABC, whose tripolar coordinates are in given ratios; their counter-points (twin points) are on a diameter of (H), counter point . conic of t (4 ). The orthopole w of t is the midpoint of the distance of the twin points, counter points of two points on t having their tripolar coordinatesin given ratios('). 30. Generalization of Hamilton's theorem((I). Let again P be any point on t, and PaPbPc its pedal triangle, 'and consider the circle (y), image of the circle AP3PP, through BmCm; . ('Y) passes through a, since the image of w through BmCm is the projection m of A on 't (section, 27) ;. further (N)passes through Ap, and also through the image pn of .P through B.0,.; hence (y) has 1'"Ah as diameter and passes through P,, (Fig. 6). Using the known property

(1) For a special case, see: Meytelon, JV, 1913-1914, p. 92. (2) On the Lemoine point of .TaIbIc,see: Gal latly, M, fern Geometryof the triangle, 2's-ed., p. 90; Thebault , NA, 1916, p. 531, 1918, p. 232. (3 ) Meytelon, JV, 1913-1914,,p.,92. (4) Nenbarn Projectineg et contreprejedions d'trn triangle. MB. 1890, p. 25. . (1) Goormaghtigh,.M, 1922, p. 474. Fox a .special ciso see Thebault, M, 1922."p. 70. (e) Hamilt n's theorem is the special case whent-=10. 96' R. GOORMAGHTIGH: about the radical axes of three circles, we. find that B1300 P&P. and Paw are concurrent. If PaljbPCis the pedal triangle of 'anypointon t, th4 straight line joining Pa to thepoint (BmCm,PbPc) passesthrough the ortlaopole w of.t (1). In section 24,. we have found this theorem as a. particular case of the final result in section 11. 31. The preceding proof leads to other important theorem, giving a generalization of same known properties of the Feuorbach point. Let .Po be the point where PPa meets again the circle PaPbPc, and denote by ra the angle of AP with BC. Then, P"wPaAR being cyclic,

or (section 27)

Therefore

If P' is any point on t and if the pedal circlePaPbPc meets again PP,, at P0, then the circumdiameter t and . the parallel to Pow drawn through P are isogonal lines in the angle BP C. Noting further that, if Pa'Pb Pc' is the pedal triangle of the counter-point P' of P, PIP' and Pa'Pb' - are perpendicular to BP, OP, we find the following theorem:. If P' is the counterpoint of any point on t, and Pa'Pb Pc' its pedal triangle, then wPa' and the perpendicular from Pa' on t are isogonal con- jvgate8 in the angle Pb Pa'.P,, . Or, the orthopole of a circumdiameter t is, in the pedal triangle of the counter-point of any point on t, the focus of an inscribed parabola having its direetrix parallel to t (2 ). 32. Generalization of Mannheim's theorem(g). Let T be the point where P'PC' meets again the circle PaPbPCPI Pb'P', T' the point where wT cuts AAA. SinceawTPa'Pa is cyclic, wT'A,~Pa is also cyclic and r' is on the circle (y). Hence TIP" is parallel to BC, and, since P" is the image of P through BmCm, AT' =PPa =-rP' ; therefore w r passes through the ' mid-point of 'AP'.

(1) Fontend, NA, 1906, p. 55; Bricard, NA, 1900,p.59-,* Goormaglitigh, JV, 1913-1914,p. 69; Th6bault, NA, 1916,.p. 497. (2) - For this theorem, in anQther form, see Fontone NA, 1907, p. 288, and a more elaborate proof by Bouvaisy, NA, 1913,p. 137. (3) Goormaghtigh, JV, 1913-1914, p. 69; Th6bault, TMJ, 1921; p. 36, M, 1923, p. 63. Mannhein's theorem "is thespecial case for t=IO. THE ORTHOPOLE. 97 If the counterpoint of any 'point P on t,' andif thepea al circle Pa'1'b'Po' of l ' meets.again Pa'P' at r; thenthe straight line joining T. to the midpoint of Al" passes throughthe orthopolevi of 1. 33.. Let now ' Va,, V, Vo' be' the points-where 9 'Meets'BC, CA, AB, Va', Vb', Vo' the images of Va, Vb, Vothrough the circumcen'tre, 0, 17'1,. Vbf', Vo'' the counter-points of Pat,VW, VO''; By as known property, AVa', BVb', CVOlare concurrent, the point of concurrence being on the circle ABC; therefore 'AVa", BVb",CVO" are parallel. If Va' is the' point P of the' preceding section, ' then TP' is equal to twice the distance OA., and the join of r tothe mid-point of AP' passes through H. Denote by Va, Vo Vo the points where t' meetsBC, CA, AB, by Va', 1,to Vo!the images of Va; Vb, Vothrough 0, lay Va", pbrr, VO"tlae counter- points of Va', Vb','V O'; AVa", B.Vb", 0Vo" are then parallel -and their mid- paints are on a straight line passing through the orthocentre of ABC and through the.orthopole' of t (1). 34. The envelope of the reciprocal line t' of the circumdiameter' t is the inscribed conic having 0 and H a's foci; the projection H' of H on t' is on the nine-point' circle and (H', ....) (W .... ). I'Ience, as w-r H' when t passes through A, B,C , A,,,,. B,,, C,,,, .H' is always W.' The ortlaopolegar of the circumdiametert is the projectionof H on the reciprocalline of 1(.2). 35. Denote again by Vv and, V, the points wheret meets AC' and A B and suppose that, AB and AC remaining unchanged, 0 slides along t; then B. Q. envelops a parabola tangent ,to AB and AC and therefore-the ' envelope' of BC is also aparabola tangent to AB and AC. Hence (section 10) the orthopoles of t in all the triangles AB C .are on a straight line 1,. But when B Vo, the, orthopole is the projection VOAof Voon AC, and similarly the projection V,, of TVi'on AB is also one of, the orthopoles; hence the locus h of these orthopoles is Vb0V~b. If Va, Vb, Vaare thepoints wherea circumdiametert cuts the sides of ABC, and if, Va,, Vao,Vbo, tb~, V.)-,-Kb arethe projections of Vaon' AC and AB, of Vbon BA and B C, of V, on CB and CA, Men V,,V0,, Voa TV., Vbaintersect at the orthopoleto of t (3 ).

(1) goormaghtigh ET. 1914, p. 555, NA, 1917, p.199. (2) MeyteIon, JV, 1911-1912,p. 8; depiney, M, 1922,p., 313. (3) JV, 1913-i914,p. 104. 98. R.GOORMAGHTIGH:

Using further the generalization of Hamilton's theorem (section 30) for the' points Vb and V,,, the following theorem will easily be obtained : The five ,points Vb , Va,, (Ve Veal AftON) (Yb Vim, 4tBm), A' are on a straight line (1). 36. We will now consider the results'of section23 in thespecial case when .P is the circumcentre. Call q. the orthopole of the E u l e r line OH, ƒÓ1 the homologous point of gyp, in AmBMCm, It is known, from section .23, that the double point of the obtained limagon is the image of tp, through the mid-point of 0g0 0Hence is the image of O,, through ƒÓl'. If we note that 0 isthe orthocentro of mBmCm and that the orthopole w of circum diameter t is •the focusA of a parabola inscribed in A.Brn C,,, having t as directrix, we find the followinng property : The lo6us of the images of the foci of the parabolas inscribed in a triangle, through their directrices, is a limagon; this curve passes through the. vertices and is a conchoid of a circle equal to the circumcircle having its centre at the orthocenter of the given triangle. The double point is the image of the circumcentre through the. orthopole of the Euler line (2 ).

IV. Analytical notes on the orthopoles of the circumdiameters.

37. It has been proved, in section 27, that thedistances wAm,

WBm, wCm are as cos ƒÆa, cosƒÆb,' cosƒÆc, hencethe normal coordinates of win AmBmCm are as see ƒÆa., see ƒÆb, seeƒÆc For instance, when t=I0, wis the Feuerbach point q (section, 43) and the normal coordinates are as 1/(b-c), 1/(c-a), 1/(a-b); when t is the Eu1er line OH, w is the point q ,whose normal coor- dinatesare as a/(b2--c2),b/(c2--a2), cf(a2--b2). 38. The following theorem is also useful. . If.be the barycentric tangential coordinates of a straight line z , in ABC, the orthopole w of the circumdiameter perpendicularto r is the focus of the parabola inscribed in ,AmBmCmand having its axis parallel to T; hence to is the counter-point in ' AmBmCm of the point, at oo on . If be thebarycentric tangential coordinates ofa straight line in ABC, the A,2B„yC„1nornca1 coordinates of the orthopole-of the circumdiameter perpenaiczUiciriu arc (1) I3lanc, BME, 1.898,pr 79; Goormaghtigh, JV, 1913-1914,p. 100; Thebault, M, 1922,p.- 404. (2) Goormaghtigh, NA, 1918,p. 246. THE ORTHOPOLE 99

For exaniple', when t•ßOK, perpendicular to whose barycentric coordinates are as w is the point whose AmBmCm, normal coordinates are as 1 /a(b'-- c2), ..... i. e. the .Steiner point of AmBnCm. Using the preceding expressions for the coordinates, 'the following theorem will easily be proved : mais on the conicpawing through .AmBmCm and having its 'tangents at those points parallel to the isogonal lines of the joins of .A; B, C to the points, where a perpendicular to t meets BC, CA, AB (1). 39. Let. now be the normal coordinates in AmBmCmof any point A. Then WA meets the nine-point circle again at the point whose AmBmCm,normal coordinates are

Let w le the orthopole of the circumdiameter perpendicular to the

straight line whose ABC barycentric tangential coordinates are as , and A a point whose AmBmCm normal coordinates are then

Aw meets the nine point circle again at the orthopole of the eircumdiameter perpendicular to the straight line whose normaltangential coordinates in ABC are as

40. Consider, on a circumdiameter t, a. point Pwhose ABC barycentric coordinates are as a, a, y and let PI be the counter-point

(a2/ƒ¿, b2/ƒÀ, c2/ƒÁ) and the homologous point of P•L in AmBmCm; the ABC barycentric coordinates' of Pl' are . then as , Hence, if

the ABC equation to pp,, will be . Further the ABC barycentric coordinates of the •‡ point on the polar of P for the 'cireum - circle are as ƒ¿ƒ¿',ƒÀƒÀ ',ƒÁƒÁ ' . Hence the barycentric tangential equationn to the parabola in ABC is and the polar of P (

is. Therefore the reciprocal line of PP ƒ¿ ƒÀƒÁ) II in ABC is the polar of P' for the parabola; but, beingon the direetrix t of (1) Goormaglitigh, M, 1923, pp. 48, 282. (2) Goormaghtigh, M, 1923, p. 205. For a special case , see Thebaul t, Soc . Sclentifque tie Bruxelles, 1922, p. 323. 100 R. GOORMAGHTIGH : the polar of P passes through the focus w and is perpendicular to Pw. If P is any point on the circumdiametert, P' the counterpoint of P in ABC, Plr the homologouspoint of P', in . A,nBm C,,,, the reciprocal line of PPI' in ABC passes throughthe orthopdew of t and is.perpendicular to Ptrs(1). 41. By a known theorem, when P is a point on the seventeenpoint cubic (2), P, G and PI are on a straight line. If P is one of the points wherea circumdiametermeets the seventeenpoint cubic, the reciprocal line of GP passes throughthe orthopolew of t, and is perpendicular. to Pw(3 ). For- in.atanc : The reciprocal line of IG is tangent to the incircle at the Feueriae pole of the Euler line OH is the projectionp,orthopole of 10;the ortho point of G on the reciprocal line of GK; the orthopole of the Brocard diameter OK is the projection of G 'on the reciprocal. line of GK. 42. An interesting problem, suggested by the preceding article, may be mentioned here. When P is a point on theseventeen-point cubic, the reciprocal line of GP meets the nine-point circle in W, orthopole of OP, and in. another point w', orthopole of OP', PI being the counter-point of 'P, and it is known(4) thatthe polars of P and P', for the triangle ABC, are parallel to the polars of P and P for the circle ABC. This remark suggests the following problem: Consider two circumdiameters t and t' perpendicular to the pdars of two counter-points P and FI for the triangle ABC. If the ortliopole ar of t is the pr(yection of P on the straight line Joining w to the orthopole wl of t', then the locus of P is a curve of thefourth order composed of the seventeenpoint cubic and the radical axis of the circum-and ninepoint Circles(5) . Let be the ABC barycentric coordinates of P and P: their ' polars for ABC are

Hence the normal coordinates of w': in AnaB,n C,n are as (1)Thebault,TNJ 1921,p.M,1923,p.65 . (2) Locus of the foci of the inscribed tonics whose focal axes pass through 0. (3) Goormaghtigh, M, 1922,p. 164; 1923, p. 269. , (4) Goormaghtigh, M, 1914, p. 12. (5) Goormaghtigh, M, 1922,p. 208, 1923,p. 228, IM, 1922,p. 39. THE ORTHOPOLE. 101

and the image of tar' through 0, is the orthopole of the diameter el perpendicular to the straight line whose ABCbarycentric equation is (section 39),

But the normal coordinates of P are'

Hence Pas meets the circle ~1„LB1,C„a in another point, orthopole of the eircumdiameter t"1 perpendicular to the straight line whose ABC barycentric equation is

Now 9' and t" must be parallel; the condition is

V. Feuerbach point and other special orthopoles(1).

43. The Feuerbach point ƒ¿ƒÀƒÁƒÂƒÃƒÄƒÅƒÆƒÓis the orthopole of the circunidia-

meter 10 passing through the incentre ; the theorem of section 26

shows that the pedal circle of I touches the nine-point circle at c... The Feuerbach point is the point of contact of the in-circle and the

nine-point circle C). Further, q3 is the centreof the circumscribed. rectai-

(1) A good many of the properties of the orthepoles of. the circumdiameters have been obtained by generalization of theorems on the Feuerbach point, and all the results we mention in the following sections can easily be found by direct methods; to avoid overloading the chapter, these proofs have been omitted. (2) K.W. Feuerbach (1800-1834) has published in1822 his work Eigenschaften einiger merktatrdiger Ptnkte des geradlinigen Dret8oks, containing the theorem on the contact of the incircle and nine-point circle. In his proof, he shows that OqI= in-r. Proofs of Feuerbach's theorem: Mention, NA, 1850; Casey, QJ, 1862; Weill, NA, 18W, p. 259; Lignieres, •JME, 1880, p. 2; Casey, Irish Academy, 1886; Godt, AMPh, 1886, p. 436; Mac Cay, . Irish Academy, 1889; Milne, JME, 1890, Volp. 2; 71, Lauvernay,p. 387; Go'dt,nehener JME, 1890, p. Berichte,493; Vautr6,1896, p.119;Mannheim,N4,1903,JME, 1895, p. 93; Lappe, Crelle, p.13;NA,1905,p.. Fontene, NA, 1907, p.158257;Fontene,NA,1905,260; Bouvaist, NA, 1906, p. 510 Pellet,NA,p.271;1915, 1907 p.155;Thebault, p.72;M,1924, NA,1910 p. 63. F or bibliographical notes, see J. S. Mackay(Edinburgh Proceedings,, v. 11), Max S imon (Fnttoicklung der Eiementar aeometrie in XIX Jahrhundert ), L a n g e (Oeschichte des Feuerbaehechen .Rietses), Non berg, M, 1922, p. 353. See also IM, 1916, pp. 78, 238, 271; 1916, p. 24. 102 R. GOORMAGHTIGH ; gular hyperbola passing through I (Feuerbach's hyperbola)(section29); the joins of D to (Bm Cm, EF),of E to (CmAm, FD), of F to (Am Bm, DE) intersect at (Hamilton's theorem)(section 30) (1);if D•L, E•L, F•L are the images 'of D, E, F through I, the straight lines joining D•L, E•L, F•L to the mid-points of AI, BI, CI intersect at (Mannhei in's theorem)(section 32) (2) ; the common tangent at to the incircle and the nine-point circle is the reciprocal line of 1G (section 41)(3); the Am Bm Cm normal coordinates of are 1/(b-c). 1/(c-a),1/(a-b) (section 37); hence the ABC barycentric coordinates are (b-c)2(s-a),.... ; the circle Da Eb Fb passes through (special case of Lemoyne's theorem, P being the image of I through 0); is also on the pedal circle of the Nagel point N (the counter-point of N being on IO) and on the circle passing through the points where AI,BI,CI meet the sides (special case of section 29).

44. Call a•L, b•L, c•L the sides of DEF; then

But the barycentric coordinates of in DEF are (section 6)

Hence is, in DEF, the counter-point of the point at •‡ on a

perpendicular to the Eu1er line of the triangleDEF; but, by a known property, IO is the Euler line of DEF. The Feuerbach point is the focus of a parabola (ƒÎ) inscribed in DEF and having IO as directrix(5). Hence the two triangles Am Bm Cm, DEF are circumscribed to a (1) Salmon, Conies;Gerono, NA, 1865, p. 220; Mannheim. NA, 1903,p. 106; Thebault, M, 1913,p. 180; NA, 1914,p. 106; JV, 1913-1914,p. 37; Goorma- ghtigh, JV, 1913-1914,p.69. The points (Bm Cm,EF),....are called Hamilton's points. It will be further proved that Am,Bm, Cw, Ah,Bh,Ch, are on a conic; using a known property about the points D, E, F, where the conic cuts the incircle, we find that the Hamilton point (Bm,Cm,EF)is also on the line joininj the pointswhere the incircle meetsEBm FCm (Servais, M, 1915,p. 137). (2) Mannheim, Bull. sciencesmath., 1902, p. 95; Thebault, M, 1913, p. 201; Goormaghtigh, JV, 1913-1914,p. 69. (3) Aubert, JV, 1911-1912,p. 112. It followsalso that the lines joining so to (BC, AI), to Am,to D and the tangent to the incircle at are harmonic (Rouquie Malgonzou, JV, 1911-1912,p. 72). Further, the tangent to the incircle at is tangent to Steiner's inscribed ellipse (Lemoi ne, JME, 1891, p. 143). (4)Hendle, Goormaghtigh, IM, 1918, p. 115. (5) Micha1, Journ. math.spec., 1895, p. 11; Th ebault, JV, 1911-1912, p. 1; M, 1912, p. 194; 1913, p. 180; NA, 1914, p. 106. On this parabola see also Cristescu, M, 1925, p. 260. THE ORTHOPOLE, 103 parabola having q as focus and 10 as directrix, The parabola is Kieper is parabola in DEF. In a triangle ,ABC the focus of Ki ep e rt's parabola is the Feue r b ac la point of the triangle formed by the tangents atA,B, C to the cireumcircle (1). 45. Several known theorems concerning the Feuerbach point can be-deduced from; the preceding section ' and the geometry 'of the parabola (Fig. 7) .

The images of Io through the sides of z)EF passthrough ƒÓ; the

images of p through the sides of DEF are on IO and have the same

centre of gravity as D, E, F(2 ).

The Feuerbach point is the point of concurrenceof the isogonal lines,

in -the angles D, E, F,. to the perpendiculars from D, E, P on 10(3 ).

Hence, if we note that the perpendicular - fromD on EP is isogonal to DI in the angle D , and is parallel to 41, we find that

(1) Goormaghtigh, M, 1913, p.'1711 1915, p. 11. Hence, since the hiepert parabola touches the Lemoine axis, the image ofthe Feuerbach point through the Leinoine axis is on the Eu•ler line (Goormaghtigh, , M, 1915, p. 11). . (2 )Mac Vicker, 31, 1914, p. 132; in this theorem AmJ),nCm may be substituted to DEF and the projection of G on 10 to the cntroiu of D.EF (Goormrghtigigh M, 1914, p. 133). (3) JV, 1909-1910, p. 160. 104 R. GOORMAGHTIGH : the angles of g)D,-q)E,q 7 (Hamilton's lines) with the sides of ABC are respectively equal ' to the -anglesof 01 with the bisectrices AI,-Br, CI(1). It follows that cD, q)E, qF ,pass through the projections of the points where 10, meets. the sides on the internal bisecttricesof the angles of the triangle AB C(2). It -may also be noted that D, on the circle 'I* being the homo. logous to a point of the circle 0, where the tangent is parallel to AmA1,Amq)D = Dq)Ah. The straight lines cD, cE, pF are bisectrices of the angles formed by cpAmand cAh, cBm. and qBh, q)Cm and qi Ch(8 ). Further, the perpendiculars drawn to EF, PD, DEthrough the pink where they meet 10, are tangents to (ir) and therefore form a triangle zviaosecircumcircie passes through 46. To determine the contact-points of with EF, PD, DE, we note that AmgAh=AmB,fA, B C. Now if D1 is the point where (pA,n meets. the incircle. hence DD, is parallel to EP(5). But the orthocentre h of REP is on 10, and its image h' through EF is on the nine-point circle ; therefore, DD, being parallel to EP and to D'h', the straight line la'q), image of IO through EF, is perpendicularto Dqq. Hence the contact-point of EF with lies on cpD1. The contactpaints of EF, PD, DE with the parabola lie on pA... Spl3m,q) Cm respectively( It follows also that the images of I0 through ET, FD, DE pass' through the midpoints of AH, BR, CH, since these points are the images of Am, B., Cm through 0,(7). 47. It nay further be noted that, if a parallelis drawn through A to 10 and if the isogonal conjugate of the parallel meets DI at .D2, then ID =B, since the triangles IAO, AID2 are equal; but 9 is on .the isogonal conjugate in BmAmCmof the perpendicular from A, (1) Goorm'aghtigh, M, 1915,p. 13. (2) Goormaghtigh, M, 1915,p. 13. (3) Rouquie-Malgouzou, JV, 19101911,p. '36. (4) Sevrin, M, 1923,p. 93. (5) 8ervais, M, 1915,p. 132. (6 ) Br1card,. NA, 196, p. 90; Th6bault, JV, 1911-1912,p. 1; M, 1913,p. 180;. NA, 1914,p. 106; Servais, M, 1915,.p. 182. These contact-pointsare also on Bh("I,....(Servais, M, 1915,p. 136). (7) Rouquie-Malgouzou, JV, 1913-1914,p. 52. THE ORTHOPOLE, 106 to I0, and AD2 is therefore parallel to the straight line joining 9 to the mid-point of AH. But AH=2OAm : If the lines joining ƒÓ, to the midpoints .of AH, BH, CH meet DI , El FI at Ds, Es, F3, ID3, IE3, IF3 are equal tothe distances- from Am, Bm , C m to the mid-points ' of the arcs BC, CA, AB of the circumcircle(1). 48.Considering further the three circles 0, Oo ,I and using the k nown properties concerning the collinearity ofthe centres of simi - litude of a group of any three circles, we findthe following theorem : The straight lines ,joining the Feuerbach pointƒÓ, to the inner and outer centres of similitude of the circum- and in -circles pass - through the centroid G and the orthocentre H respectively (2). 49. Mannheim's lines. We, have seen (section 37) that the joins of D', E', F' to the mid-points of Al . BI CI the straight lines D'4p, E'q., Pp are calledare Mannl.eivesconcumrent at ƒÓ, lines. - - D' being on the incircle homologous to a point of the nine-point Lcircle, where the tangent is parallel to AN., D'q is a bisectrix of B,,,p Cm, and BmCm is divided by D',o in the ratio pBm :q.aCm or (c- a) :(a - - b). T he three Mannheim lines meet I G on the sides of the medial triangle(3) . It may then be noted that the join of A to (BmCm, D'p) meets BC in a point of the reciprocal line of IG, andthat the line joining this point to I is parallel to D'q and perpendicular to Dq); hence we find that the reciprocal line of IG touches.the circle I at qq (section 43). 50. Considering the triangle AB. C., and notingthat the Ha- milton point on B,,,C. divides BmCm in the ratio (c-a):(a-b), it will be easily proved that the straight line joining C to (B;,,Cm,EP) meets AB on the reciprocal line of IG, i, e. onthe tangent at lp to the incircle. The straight lines joining each. Hamilton pointto the ends of the corresponding sides n eet the sides of the triangle ABC on the tangent at P to the incircle('4 ). Hence, the sides of the triangle ABC meet the corresponding sides of the triangle formed by the Hamilton points on the tangent at p to the ninepoint circle. 51. Since the .4B, iCm barycentric coordinatesof op are (1) Mineur, M, 1922, p. 89. Vetoing, JV, 1911-1u12, p. 128; ahebault, NA, 1915, p. 198; 1916, p.Se 116; rvais, M, 1915, p. 133, Th5bault; NA, 1922, p. 402. (3) Th6bault, hi:, 1913, p. 203; de ,Lepiney, M,1922, p. 272. (4) Th6bault, M, 1913, p. 183; NA, 1914, p. 144. 106 R.GOORMAGHTIGH : .... ; and those.of the point where AAm meets again the nine-point circle are the straight line, joining those two' points meets BmCmon the straight line

The Fe iserbach point is on the lines ,joining the points where AA,,,, BB,,,, CCm meet the nine-point circle to those where B,,, Cm, C.A., AmBm respectively meet the straight line passing through the centroid 0 and through the i8otomic conjugate to I(1 ). 52. In section 46, we have already met with interesting properties concerning ƒÓAm, ƒÓBm, ƒÓm; we win nowprove the following theorem The points where the internal lrisectricesmeetMe perpendiculars from D,, E, F on 10 are on, . Denote by L the point (AI, pAm); then

hence 1, p, D, L are cyclicand LD1=IqL; DL is isogonalconjugate to 'Dip' in FDE; thereforeDL is perpendicularto 10. It may also be noted , that the perpendicular to10 drawn from the image of D through the mid-point of EF passes through the image of L through EF and therefore passes through the point (EF, tPAm),where BF touches the parabola (ir)(8 )- 53. The points cpa,'pb,q,,. By substituting Ia0, Ib0, Ic0 to I0 the preceding theorems will lead to properties of the points -of contact 9)a, pb, 'Pc of the nine-point circle and the ex-circles; these points are the orthopoles of 1. 0, Ib0, 1,0. The sine and cosine of the angle formed by 10 and BC are

and similarly for I.0

(1) de Lepfney, .M, 1922, p. 273. When two points $1, X are iso!onfc conjugate in ABC, the points (BO, A$1), (CA, B$1), (AB, CX1), are the images of (B0 A$2), (CA,B$2), (f113,Chi 7) through Am,Bm,Cm rospectively (2) Goormaghtigh, M, 1914, p. 191. (3) Thebault, 1, 1913, -1). 200; Goormaghtigh, M, 1914, p. 190; Cris- teseu, M, 1924, p. 413. THE ORTHOPOLE, 107

But Hence the sine of the angle formed by 10 and I0is

and, the angle ƒÓ0qƒÓa being twice this angle(1),

54. Coordinates of The AmBmCmnormal coordinates of q)n are (section 24)

1/ (b-c), -1/(c+a),1/(a+b)

Hence the ABC barycentric coordinates of ƒÓa, ƒÓb, ƒÓc are

-s(b-c) , (s-c) (c+a)', (s.-b) (a +.b.)2,

(s-c) (b+c)2, -s (c-a)2, (s-a) (a+b)2,

(s- b) (b +c)2, (s-a) (c + a)2, -s (a- b)2. The ABC normal equation . to q .'0 has the, form kz + y + z. =0 ; hence q qo meets BC on the externalUsectrix of the angle BA C(2 ). The above coordinates show further that Apn, Bpb, Cgioare concurrent(!), the point of concurrence being (b+ c)21(s- a), (c+ a)21(s--= b), (a + b)21(s-- c) (ABC barycentric coordinates). It may also be noted that, whenA=60°, are on astraight line (Al). 55. The orthopolesp, and q.2 of the Euler lineOH and the Bro- card diameter OK. We have already proved (section41) that c1 and c2 are respectively the projections of the centroid G and the. Lemoine point K on the -reciprocalline of GK. But q2 is the centre of the counter-point conic of OK Kieper is hyperbola (H1)-and is therefore

(22) Lemoine, NA, 1880, p. 122; see also Fontene, NA, 1905,(1)Thebault, p.'520, M, 1922, P.118;Elgnairt,IM,1915,p.47 . Gallatly, M, 1808, p. 23. . ' . (3) Leinoine, NA, 1880, p. 122; G. do Longchamps, JME, 1891, p.'108. 108 R. G0ORMAGHTIGH : on Steine is conic a tangent to DO, CA, AB at .Am,Bm, Cm, as a is the locus of the centres of the conics passing through A, B, 0, G. Hence 972is Steiner's point in AmBmCm(1). Further as is tangent to the reciprocal line 9192 of GK; hence q'1g2 is the tangent to o- at q'2. But it is known that the Tarry-point of A4B,,1C,, image of Steiner's point 92 , through OQ is on the straight line joining. the centroid G, to the centre w' of the Br o card circle of A.B.Cm( 2 ); therefore 091, perpendicular at qi on grips, passes through w': We will now prove that g91v2is the polar of G for the B r o c a r d circle w'. By a known property, AmBmCmand its first Brocard triangle are similar and have the same centroidG ; therefore 0&1: The triangles V1w'Og,and Ogw'G are sirnilarr and

The orthopolep, of the Enllcr line and the centroid G are inverse points in the Brocard circle of A.B,,, 0m; p, i8 the 'projectionof G on the tangent. at the Steine r point cp2(f AmBmCm to the maximumcircumscribed conic of AJB1Cm. It may also be noted that Kieper is parabola i n the triangle A,nfmCm has OH as directrix and p1 as focus. Hence 9i9a perpendieu_ lar at pt on Gpl, is the polar of G in Ki ep er is parabola of A,,BmCm. 56.Morley's theoremon the Feuerb ac h point. If, through 'the contact-points D, E, F of JW, CA, AB with the incircle, parallels are drawn to El) FD, DE the Feuerbach point P of ABC is the orthopole of the F u ler line in the' triangle formed by those parallels, as it is known that 10 is the Eu1er line of DEF . Hence the results of section 55 lead to the following theorem: The Feue r hac h point 91 in ABC and the centroid of D.EE are inversepoints in the Brocard circle of DEF; c is also the projection of this centroid on the tangent at the Steiner point of DEF to the maximumcircumscribed ellipse of DEF(3 ). 57. Otherproperties of p1 and 92. Some interesting properties of 'p and 4p2are connectedwith the points ..Citi(BrCm, BA Ch) , Bi(CmAm,. A ,A), Ci(A4B,n,AX) and K ie p ert's hyperbola(H1). Since H and G are two points, on (HI), the triangles AhBkOh, A mBnCm are both conjugate to (Hl) ; therefore B1 and C, are the (1)For theoremsof sections55 and 66, see Goormaghtigh, M, 1923,p. 257. (2) Brocard, AcWemie Monipdiier, 1883. (3) -.Morley,M, 1894,p..32; Lienard, Goormaghtigh, M, 1913,p. 269. THE ORTHOPOLE. 109 poles of AC and AB, and B, C1 is the polar of A. Hence, B, C, passes through A. CIA, through. B, A1B1through C(1). Further, since A,, and At are the poles of~Bj,.C,, and BC , pi is, i n Kiepert's hyperbola, the pole of the radicalaxis of the nine-paint circle and the eircumcircle. Then, A,,,AIbeing the polar of the point at o0on BC, A.A1 passes through p2 centre of (B;); AM'Alt BmB1,C;:C,' intersect at gas; hence A1,B,, C, are, in A.mBmC,n, the contact-pointsof B'i epe r t'a para-. bolawith the sides. But p2 being on the nine-point circle O and on o', A1B1C1 is conjugateto' O,,and to aa; therefore the point whore B101meets BC is on the tangent plpx to a. The tang' Was cd - to . passesthrough (BBC,,150), (0141, ('A), (A1B1,AB) ; pip, being the reciprocalline of GS, the points (B1C,, B,mCm),....are on GK(2) , and GK is the polar 'of p$ in the Kiepart's parabola'of AmBmCm. 58.; Otherproperties of the tangentsto the nine-paintcircle at q, "pa, Pb,P6. These tangents are the reciprocallinesof GI, GIa, Gib,-GI,,; hence they are also tangent to o. The triangleformed by the diagonalsof .the quadrilateralof the four tangents is conjugateto O, and o and is .thereforeA1B101; using the resultsof the preceding section,we find the followingtheorems : The:vertices of the quadrilateralformed by the tangentsat q, p,„ pb, pa to the nine-pointcircle are twoby two on straightlines with A; B, C ; the diagonalsof this quadrilateralmeet the sideson a straightline passing through.p, and q?,. Now, 9poqi6 o has the samediagonals as the quadrilateralformed by the tangents to the nine-pointcircle at p,-3a", spa and pap,,meet at At, ppb and 9~cpameet atB1, pop.and pa9, meetat C1. 59. A few interestingtheorems will also be obtained -by considering specialcases in section39 (s) :

The straight line G9 passes through the orthopoleof the circumdia- nneterperpendicular to IS. (1) It maybe notedthat BmA,,:OmA1=(a2-a2) : (a2-b1); hence AA.,, 13B17 001 are perpendicularto Off. (2) For the theoremsof sections57 and 58,,see Deaux, M, 1922,'p. 282. (3) Goormaghtigh, M, 1923,p. 205. 110 R. GOORMAGHTIGH:

The straight line Gp1 passes through the ortliopole of the circumdiameter parcel to the Brocard, line.

The points where AA., BB,,,,00. meetthe nine-pointcircle are . the orthopoles of the circumdiametersperpendicular to the symmedians. VI.. Theorems on the orthopoles of the circumdiameters obtained by a method of generalization(1). 60. On any circumdiameter t there are two points T ,1y (besides 0) whose pedal triangles are homologic to ABC; the points of con- iao ui iue joins of A, B, (I to the corresponding vertices of these pedal triangles will be denoted by J and J2(2).Project the. figure into a new figure in which the projection of -1, is the Gergonno point of the triangle, projection of ABC; usingthe generalization of Hamilton's theorem (section 30), we find that the orthopole m' of t is projected into the Feuerbach point of the new trianglo... If we further note that the Nage 1 point is, in the medial triangle, the homologous point to the incentro and if we use the theorem on the reciprocal line of 10 (section 43), we find thefollowing property: The reciprocallines of GJ and. GJ; pass through c and are the tangentsin this point to the conicstangent tothe sides atlhe projectionsof A and 4. The centres of these conics are the homologouspoints J"2 and J'1 to JZ and J in the medial triangle. As there are only two straight lines, passing through G, whose reciprocal lines pass through rr, the theorem of section 41 becomes : The ortliopotecu is the projection, on the reciprocal line of GJ (or. GJ2) of the point where GJ - (or 0J2) meets t. Consider again the two conies tangent to the sides of ABC at the projections of Il, 42; then (section 50) the tangents at w to the conies pass through the points where the sides of ABC meet the corresponding (1) Goormaghtigh, JV, 1913-1914,p. 109; M, 1923,500; Thebault, M, 1922,322. (2) The loci of Ii, r2 and Jr.,,J2 are two cttbics(Darboux's cubicand Lucas' cubic);Darboux, NA, 1866,p. 95; Ed. Lucas, NA, 1876,p. 240; NOM, 1876,p. 94; De vu l f, NA, 1870,p. 550; Laguerre, NA,1879,p. 144; Moret-Blanc, NA, 1881 , p. 520;Schoute-Neuberg, AFAS,1891, p. 108. THE ORTHOPOLE. 111 sides of the,triangle formed by the points of intersectionof the sides of the medial triangle and the pedal trianglesof Ii and I2(1). 61. project now the figure ABC 1112JJ so .thatthe projection of Ji be the orthocentre of 'the new triangle; using again the generalization of Hamilton's theorem, we find that the projection of tar is the' orthopole of the Euler line in the projection of ABC. The coniespassing throughthe midpoints of the sidesand throughthe projectionsof I1,'12on the sides pass through.w( 2 ). But, in the projection, the circumscribed rectangular hyperbola, whose centre is the orthopole of the Euler linepasses through the orthocentre and its homologous point in the medial triangle; hence, the counterpoint conic of t passesthrough J, J2 and throughthe homologous voints J7,. J', to J,, J, in the medial triangle. But w is the centre of the counter-point conic of t : The straight line-Joiningthe centroidto the midpoint of JJ2 passes through w. Referring to theorems of section 55, we find that the tangentsfrom w to Steiner's inscribed ellipseare the reciprocallines of GJi and GJ2, the points of contact k1 and k2 being the polesof GJ and G ''in the parabolas conjugate to ABC and whose axes are parallel to the triangular polars of J2 and J1. In the projection of ABC, the Lomoine point is the projection of Xs; hence the parallels to Gtr drawn throughJ1 and J12 pass through k1 and k2. Further, as the orthopole of the Euler line. isthe centre of the circumscribed rectangular hyperbola passing through the centroid, k1 and k2 are the centres of the circumscribed cotits passing through G and respectivelythrough . J2 and J. Finally, the reciprocal lines of GJ and GJ2 arethe radical axes, of the nine-point circle and the pedal circles of IZ and I1 (from preceding result and F a u r e's theorem). 62. Special theorems. The preceding properties applied to special circumdiameters give come interesting theorems The conic tangent to BC, CA, AB al Da, Eb, F, passes through the. Feuerbac la point p and is tangent at q' to the.reciprocal tine of GI ; the normal at p passes through the point (Gf',10)(3 ).

(1) Th6bault, M, 1923, p. 48. (2) For the special case of the .Fouerbach point, see JV, 1913-1914, p. 104; do Lepiney; M, 1914, p. 249; Servais, M, 1915, p. 136. (3) Nouberg, M, 1922, p. 126. 112 R.GOORMAGHTIGH:

The conic tangent to. BC, CA, AB at A,,, B,,, C,, passes through, the ortliopole p, of the. Euler line. The conic tangent to BC, CA, AB at the images of Ah, Bh, Ch, through a Am,Bm, Cm passes through vi and is tangent at q, to the - reciprocal line of OB. The conic AmBmCmDa b.F passes tlarovghp ; the straigM line Joining G to the midpoint of ['N passes through gyp. It is easy to prove that the homologous point Ito in AmBmCm, is on the seventeen-point cubic; hence the orthopole of Or' is on the reciprocal line of Gr.

VII. Some orthopole theorems in connexion with conics. 63. Let

be the cartesian coordinates of three points A,B, C on an ellipse (.0). The, coordinates of the inid-points A.,'Bm, Cm of the sides are then

and

are the ' equations to the perpendiculars to AmCm,AmBr through the projections of Bm and C„n on the axis 2p. Hence, the values of the coordinates of the orthopole M, of the axis 2p in the triangle A.B,. C,,, are

But the circumcentre 0 of ABC- has as coordinates THE ORTHOPOLE. 118

Therefore, if w1 is the centre of the conic,

Mj 'being the orthopole of the axis 2q in the triangle AmB,nC. ; the mid-point of Ow, is also the mid-point of MpMq. The orthopoles, for the medial triangle AmBmC„., of the axis of a conic, circumscribed to ABC, are on a straight line joining the centre of the conic to the centre of the circumcircie of ABC. The circumcentre divides the distance of the 'orthopdes in the ratio of the squares of the axes(1). When the conic is a rectangular hyperbola, Mp,MW coincide with the mid-point of Owl. When the conic is a parabola, the orthopoleof the axis for the triangle A,nBmCm is on the diameter of the parabola passing through the circumcentre of ABC. If, in this case y1, y2, y3 are the ordinates of A, B, C, and y2=2px the parabola, the orthopole of the axis in AmBmCm has as coordinates

and those of the circutncentre of ABC are

Hence the distance of those two points is p(2 ). 64. It may now be noted that, if GI is the projection of the controid G on any straight line,. the orthopoles of that line in AmBm0. and ABC are on a straight line passing through GI and that G' divides'the distance of those two orthopoles in. the ratio 1:2('). Hence

(1) Goormaghtigh, NA, 1919, p. 137. (2) Thgbault, M, 1923, 'p. 202. (3) Se rvais, M,. 1923, p. 340. 114 R. GOORMAGHTIGH : it is possible to deduce from the preceding results properties of the orthopoles of the axes of circumscribed conics,the orthopoles being taken in ABC. An interesting case is obtained when the centreof the conic is the centroid G: In a triangle, the orthopoles of the axes of Steiner's circumscribed ellipse, are on the Eul er line and the mid-point of their distance is the midpoint of GH(1). Since the orthopolar lines of the points on theEu1er-line envelop the inscribed parabola having that line as direotrix (section 18), i. e. Kieper is parabola, we find the ' following theorem : The axes of Steiner's maximum circumscribed ellipse are tangent to .Ki epe rt's parabola(2) ; . 5. A few interesting6 results(3) will be obtained by considering triangles in connexion with a parabola y' = 2px; calculations are very easy and will be generally omitted. Let y1,y2, y3 be the ordinates of three pointsA, B, 0 on the curve ; the orthopole of the vertex-tangent in ABC is (-2p, yl + y2 +VI). The orthopolesof the vertex-tangent of a parabola, fur all inscribed triangles, are on a straight 'line. When the normnalsat the vertices of the inscribed triangle are concurrent, the orthopoleof the vertex-tangent is a fixed point on the axis. The cartesian coordinates of the orthopole of the axis in ABC are

when the normals at A, B, C are concurrent in W(ƒ¿,ƒÀ),

hence the coordinates of the orthopole are (M-p) and - .2 R. If, from any point W the normals WA, WB, WVCare drawn to a parabola, and if M is the orthopoleof the axisof the parabola in the triangle

(1) Goormaghtigh, M, 1922, p. 498; Sevrin; 1923,'p. 279. (1) Goormaghtigh, M, 1922, p. 498. (3) Goormaghtigh, WT, 1918-1919, p. 32. THE ORPHOPLOF. 115 ABC, the projection of War on the axis is p, and WM is divided by the axis. in the -ratio 2 : 1. When W moves on a straight line, the locus of M is also a straight line, For the triangle formed by the tangents at A,B,C the,coordinates of the orthopoles of the vortex-tangent mid theaxis are ..

By means of these expressions, several theoremsof the preceding sections and other properties will easily be proved; it may be noted, for instance, that, if the normals at A, B, C, are concurrent at W, the orthopole of the axis, in the triangle formed by the tangents at A, B, C is the projection of W on the vertex-tangent. Further the vertex-tangent and the directrix have the same orthopoles in all inscribed triangles whose centroids are on a given iii nwly (2). Consider now the triangle formed by the normalsat A, B, C ; the, orthopoles of the vertex-tangent and the axis are then

and

The orthop~oleof the axis in the triangle formed.by the tangentsat A B, 0 and the orthopoleof the vertex-tangentinthe triangle formed by the normals at A, B, C are on the same diameter of the parabola. The distanceof theorthopoles of the vErtext-angentin those two triangle is seenfrom the vertex of the parabola at a right angle. We note finally that if the normals at A, B, V are concurrent at W (ot,0), the coordinates' of the vertex-tangent and the axis, in the triangle formed by the curvature centres atA, B, C are

and

(1) When, in a quadrilateral formed .by the -four, straight lines a, b, c, d, one of these lines a is parallel to the line joining a fixedpoint on the axis of the inscribed parabola to the orthopole of this axis in the triangle b, c,d,the sameproperty exists for each of the lines b, e, d (doormaghtigh, 1M, 1919, p. 13).' . (2) Servais, M, 1923, p. 340. 116 R.000RMAGHTIGI1 VIII. The isopole. 66. When, from the vertices of a triangle ABC straight tines are drawn making.pith a given straight line m equal angles at al, /91, y1,the straight lines drawn from oil,/31; 'y .and making angles equal to 0 with BC, CA, All respectively,intersect at a point Me, isopoleof m for the triangle ABC and the angle 0('). We will prove this theorem by a method showing that the isopole theory can be deduced directly from the properties of the orthopole

Let 0 be the circumcentre of ABC, 00e the perpendicular at 0 onA 0, A0ea straightline making the angle 2 --8with OA ; the circle (Osi OeA) meets AB and AC at Bo and Co (Fig 8).

Q being any point on the circle ABC, Q,Qe is the perpendicular at Q on AQ and AQe the. straight line making with AQ the angle (1) Third, ;,ProweedingsB1inburgh Much. Soc., 1913, p. 17;N e u b e r g,. M, 1914,p. 89; iiervais, M, 1922,p. 203; Th6bau1t, NA, 1918,p. 219;N.A, 1918, p, 210; M, 1922,pp. 205,411; 1923,p. 268. THE ORTHOPOLE. 117

(AC),AQe) equal to then Qe is on the circle ABoCo and thQ Simson line for the angle 0 of Q in ABC is the ordinary Simson line of Qe in ABo Co. Hence BoCo is in ABC the S i m s o n .,line 'for , the angle 0 of the image of A through 0 and therefore makes the angle 2 -0 withBC. To prove . the isopole theorem, note that, if mmoves parallel to itself, the figure a,i91y1112eremains unchangedin. shape and size, Me sliding along a straight line.parallel to Acct,BR1 i Cy,; consider the case when. m passes through B, the points and lines being all denoted by the same letters as in the general case; lotBe be the point . corresponding to B, like Qe corresponds to Q. The perpendicular me at al on Act, passes through Be, since ABe is a circumdiameter in the. circle ABBeotl.

The strsaiorht line drawn through ƒ¿1 and making the orthopole of mƒÆ

BC is perpendicular to BoCo and passes through the orthopole of me in the triangle. ABoCo; the straight line drawnthrough B•ßƒÀ1, and making the angle 0 with CA is the ordinary Simson line of Be , for the triangle ABo Oo; hence 111e is the orthopole of me in the triangle ABoCo.

Returning to the general case, we find the following theorem :

If me is the perpendicular at al on Aa1 and if B0C0 is the Si m so n line, for the angle 0, of the image of A through the circumcentre 0 (Bo on

AB, Co on AC), then the isopole Me of m for theangle 0 in ABC is the orthae pole of me in ABo C,,.

67. The isopole He of m in the triangle ABC is on the ordinary

Simson lines of the points where -no meets the circle ABoCo; but these points correspond (preceding section) to the points where m cuts the circle ABC. ,

lies on MƒÆthe Simson lines, for the angle 0, of the points where m meets the circumcircle ABC(1).

Similarly several properties of the orthopole will give theorems about the, isopole(2) :

When m passes through a fixed point P, the locus of Me - is a conic circumscribed to the pedal triangle of P for the angle 0, and the isopoles of two perpendiciclar lines are on a diameter of the conic ., (section 16).

1) Th6bault ( ,'NA, 1914, p. 221. (2) For these theorems, proved by other methods, see Servaie M,1922, p. 262, p. 361; M, 1923, p. 150, p. 239. 118 R. GOORMAGHTIGH Whenm movesparallel to itself, Me slides alonga Simson line,for the angle 8, in ABC,' and the*Simson tine makesthe angle it-8 with m. WhenMe moveson a given straight line Z„ the isopolar triangle of Me envelopsa parabola and the Si m s o n line for the angle 0 of the focus of theeparabola is parallel to 11. Whenm is a eircumdianneter,the locusof MBisthe circle passing throughthe points JP,,, M,, M'', on BC, CA, ABand taken so that the angles of OM„ with B C, of OM'bwith CA, of OM,, with AB are equal to 0 (pedal circle of 0 for' the angle 0). If two 'straight lines m and ml are perpendicular and pass through a fixed point P, the lines drawn through their isopoles and making the 'angle -7r-0 with m and ml are S i Yns o n-lines for the angle 0, and as they are perpendicular, the locus of their intersection is a. circle (tritange'nt circle to the tricuspidal envelopeof these Simson lines). Further, as the mid-point of the distance of these isopoles is a fixed point, the locus of the intersection of the lines drawn through the isopoies and making the angle 7r-0 with mniandm is also a circle. The distance of the isopolesof two circumdiameterson the circle .M M'b11p,corresponds to a centre-angleequal to twice the angle between the given diameters. f theI linesdrawn through..Fl, Bs, C and makingwith a circumdiameter t the angle meet t at al, al, yl, then the circles goingthrough A and a,, .11and 01, C and yl and having respectivelytheir centreson At',,111'„ mom,,,, M1M,, pass throughthe isopoleof t (1). 68. Iii a system of triangles inscribed in a circle and circumscribed to a conic, the Simsoi lines for a given angle 0 of a fixed point on the circle pass through a fixed point;when the conic is a parabola, the Simson lines are parallel. The locusof the isopolesof a given straight line, for the angle0, in a systemof triangles inscribedin a circle and circumscribedto a conic is a circle; when the conic is -a parabola, the locus is a straight line(2). 69. The final theorem of section 9 has been generalized for the case of the isopole as follows (s) : If throughthe isopole. MBa straight line u is drawn making with to the angle lr-0 and throughA, B, C the lines ha,hb, h, making the angle 0 with B C, CA, AB, the secondtangent from 11?Bto the parabola tangent to u, ha,&, 'homakes the angle 8 with the reciprocal line of m in ABC. (1) This last theorem does not seem to-have been mentioned yet. (2) Servais, M, 1922, p. 268. (3) Servais, M, 1923, p. 152. THE ORTHOPOLE. 119

Let HQ, Hb, Ha be the points where u meets ha, hb, ho; these points are the isopoles of the parallels ma, mb, no drawn to m through A,B, 0. The reciprocallines BC, CA,A13, mr of ma,mb, m0, m are tangent to a' parabola and (BC, CA, AB, mr)T(ma,mb, me) m)7(Ha, Hb)Ha, Me). Hence the lines drawn through Ha, HbeHe, Me and making the angle 7r-9 with BC, CA, AB, mr respectivelyaretangent to ayparabola tangent to v-_ HaIIbHA. ; ,

IX. Orthopole theorems in solid geometry. 70. ABCD being a tetrahedronand a, ,3, y, S the projections of A, B, 0,D on a plane m, denoteby a', 13',y', 'S'the orthocsntresof the triangles $y8, y8a, Sc$, a,Qy and by (a), (,Q),(y), (S) and (a'), (,6f), (y'), (o') the perpendicularsdrawn from a, 8,,y, S and a', 8', y', S' on BCD, CDA, DAR. ABC. The straight, linpc (a). (8). (-vl.(S) and (a'). (Q'). (y'), (S') belongto an hyperboloid,whose centre is theintersection of theperpendiculars drawn on BCD, CAD, DAB, ABC, from the midpoints of ow', f30', yyt,

(a) is perpendicular to BCD and therefore also to BC; (S') is perpendicular to ABC and also to BC; further, aS' is perpendicular to ,8y, and also to BO, since $7 is the projection of B C on m. Hence (a), (S') and aS' are in. a plane perpendicular to BC. Similarly (a) cuts also (0'), (y') ; hence (a) cuts (R'), (y'), (S') and also (a') (at oo); the two sets of straight lines (a), (16),(y), (S) and (a'), (,9'), (y'), (S') lie on an hyperboloid, and form four pairs of parallels; hence the perpendiculars on BCD, CDA, DAB, ABC from the mid-points of oar, Mr, y7', So' are concurrent at the centre of the hyperboloid. 71. If (%,,8,,y are the projection-sof the vertices of a triangle ABC cn a straight line m (not in the plane of the triangle), (he planes perpendicular to BC, CA, AB and passing through the points dividing Aa, BO, Cy in the same ratio pass through a straight line(*-'). If x,, y,, ; (i=1) 2,3) are the Cartesian coordinates of A, B, C, m being taken as x-axis, the equation to the plane. perpendicular to BC

(1) Neuberg, JME, 1891, p. 24; AMPh, 3d ser., Vol II, p. 305. (2) de Lepiney, M, 1925, p. 191. 120 R.GOORMAGHTIGH

Similarly the equations to the planes perpendicular to CA and AB are formed by cyclic permutation of i, and each of ' them can be found by adding the'two others. 72. In a tetrahedron, thepedal spheres,of thepoints of a given straight tine cut orthogonally a 'same sphere(1). Using Cartesian coordinates it will first be easy to prove that, in a tetrahedron, the locus of the points whose pedal spheres cut orthogonally a given sphere is a surface of the fourth order. If now a straight line m meets BCD, CDA, DAB, ABC at Ua, UU,Uc, U,1, the spheres (AU.), (I3 U,), (CU,),(D Ua), pedal spheres for U a, Ua, Uo, 1h, have the same radical axis (Serret's theorem)("); and the radical axis meets 'the radical plane of the sphere (A Ua) and the pedal sphere of any point P of m at the centre M of a sphere (M) cutting orthogonally the four spheres (A U.), ,and the pedal sphere of P. Hence m belongs to the surface of the fourth order corresponding -the sphere (M). 73 If XYZT and XYZT areare two equipollent tetrahedrons to andXXV, ifVX,VY,VZ,VTYr, ZZ', TTY,the equicentre of XYZT, X?Y'Z'T' (section 6) has . as barycentric coordinates in XYZT and X Y'Z'T' the barycentric coordinates of V in I' Y7'0"Tit . Hencethe pedal tetrahedronsof all thepoints of a straight line m have a commonequicerctre( 1). X. Notes and problems. 74.. To prove that, q being the Feuorbach pointand pl - the orthopole of the E n l e r line (c< a < b)

(1) Servais, BB, 1922, p. 52; M, 1922, p. 88. (2) Serret, Geometrlede direction,pp. 266-268. (3) Neubarg, 11, 1922, p. 89. (4) G od t Acad. Munich, 1896; J V, 1909-1910,'p. 5. (5).. Th6.bault, M, 1918, p. 200; Goormaghtigl, M, 1914, p. 191; Bri.s tescu, H, 1924, p, 413. (6) Th6bault, NA, 1915,p. 142; 73ouvaist,,NA, 1917,p.,148. THE ORTHOPOLE. 121

75. Aiyais theoremon' counter points can be easily proved by using the. properties of the orthopoles of the circumdiameters. If P and F' is any pair of counter points in ABC, and if P'o is the midpoint of PP, then

The common pedal circle of P and P meets the -nine-point circle at t and w l, the orthopoles of OP and OP'. If Oa and O'a are the angles of OP and OF with BC, the power of Am inthe pedal circle is OP cos 8a•OP cos 8'a. But this rpower _ i~ w1 so a twice L11VZ,-^ 1.•.wproA of O0P'0 by the distance from Am to ww' ; therefore (section 27)

. 76. To prove Gob'stheorem(5) by meansof the propertiesof the Feuerbach point: Take on BA, CA the lengthsBA,,, CA, equal to 1W, on CB, AB the lengthsC.BB, AB,, equal to CA, on AQ, B Cthe lengthsA Ca, B C,, equalto AB. Then theeircl 9 Ma,Mo,1Y1a circumscribed to AA,Ac,BB0Ba, C6.0b are equal their radii being equal to - 10; I is their radical centre;the tangentsat A, B, C to 1bMa,If,, Maintersect on the circumcircle at thepoint whosenormal coordinates in ABC are as (b-c)-1,(c-a)-1, (a-b)-1; the lengthsAM;, and AM; are equal andA JT,,and AM, are isogonalcon, j agate. A simple proof will be obtained by using the similitude of B,,tpCan and AAA, C,,,rp4),and BaBBa,A,14gcBm and GaCC,,(o ). 77. Mac Ca y's cubic. We know, from section' 26, that, if the (1) Thbbault,NA, 1915, pp. 477478; Goorinaghtigh, NA, 1918, pp. 117, 214. (2) Thebau lt, JV, 1909-1910,p. 174. (3) Goormaghtigh,JV, 1915-1910,p. 150; M, 1925, p. 213. (4) Gall'atly,Modern Geometry of, the triangle,2&ed., p. 79;a similarproof has beengiven by Thebault, M,1924, p. 153.Other proofs : Morle y Prac. .London.Dlath. Boo., vol.. 21,. p. 140;M, 1924, p. 18;Goormaghtigh, M, 1925,p. 8.' (5) Goormaghtigh, M,1924, p. 439.. (6) Goormaghtigh,M, 1924, p..439. 122 R. GOORMAGHTIGH : counter point P' of P lies on OP, the common pedal circle of P and P' touches the nine-point circle at the orthopole of POP. Denoting by x, y, z the normal ABC coordinates of P, ahd expressing that the points (x, y, z), (11w,1ly,1/x) and ScosA, cos B, cos C) are on a straight line, we find the normal equation to the locus, of P (Mac Cay's cubic)(1). It may be interesting to note that points on the curve can be easily constructed as follows: Considerthree equal circles (radius B,) havingtheir centresat A, B, C and let A'0,B'0, C'0 he any of the common points to the circles cor- respondingto B and C, C and A, A and B ; the locusof the centre 0'0 of the cirde A'0B'0C'0 is Mac Ca y's cubic. The parpendiculars. to B'0C'o, C'0A'0, A'0B'0 at their mid-points pass through A, B, C ; ABC and A'0B'0 C'0 are orthologic and the barycentriecoordinates of 0'0 in ABC *are as the baryeentrie"nnr- dinates of 0 in A'0B'0 C'0. Hence the ' normal ,coordinates of O'o in ABC are as

therefore

Hence multiplying by x cosA and adding the three similar equations, we find the equation to Mac Cay's cubic. 78. The letters having the same meaning as in chapter VI, to prove the followingtheorem : If, throughJ and J2 straightlines are drawn havingtheir directions symmetricalto thoseof GJ1and GJ2with resp. to theSimson linesof the points wheret meetsthe circumcircle,and if thosetwo linesmeet at S, thenS i8 on the straightline joining G to theorthopole w of t, and (2). 79. If Was and are thedistances from the Fewerback point p to the points where the sides of DEF and ' AB. C. touchthe (1) Mac Cay, hish Academy, 1889. (2) Goormaglitigh, M, 1924, p. 336. THE ORTHOPOLE. 123 parabola inscribed to A„,BmC,;, and having 10 as directri.x, then (1)

80. Using theorems of sections 55 and 64, the following property will easily be proved : ED',F and ED'2F, FE 1,D and FE'2D, DF', : and DF'2E' beingthe equilateraltriangles on EF, FD, DE, the two bisectricesof each of the three angles D',DD'$, E',EE'j, F',F.F'2 are parallel to two same directions. The straight line drawn through the centroid ofDEF and having its directionsymmetrical of that of 10 with reap. to tlcosetwo directions,pass through the Fenerbach point of A.BC, throughthe Tarry point and the centre of the .Brocard circle of DEF( 2). 81. When a straight line m passes through the orthocentro H, the image M of the orthopole M of mz through mlies oil, the nine- point circle ; the straight lines joining H to the images of A, B, C through m meet the sides in three points on a straight line u, passing through .AP; M' is the projection of 0 on v1; the perpendicular from H on -u,,meets the circumcirclo in two points ; one of the' points is the focus of the inscribed parabola tangent to u, ; if through the second point a parallel is drawn to m, the reciprocal line of this parallel passes through M(3). 82. Two triangles being given itsn a plane, arethere always straight lines m having the same orthopole M inthe two triangles (4)? The perpendicular from Of on m being a Simson line in each of the triangles, m will be perpendicular to any coininon tangent 6 to the Stoinor tricusps of the two triangles. If mis any such perpendicular, its. orthopoles in the tivo,given triangles will be generally diferent and section 3 shows that if this occurs for a perpendicular. M to 0, the same will occur for any perpendicular. . , Therefore the problem requires special conditions, but pairs of (1) . Thebau It, M, 1914,p. 88. If y1f,y2,y, are the ordinatesof three points L1f L2,L;, on a parabola y2=2px having 9)as focus,and if the tangents at L,,, L2,L2 form a triangle V,, L'2,L's, then

(2) Goormaglitigh, At, 1924, p. 143. (3) Goormaghtigh, NA, 1918, p. 249. (4) thebauit, M, 1924, p. 336. 124 R. GOORMAGHTIGH: triangles solving the question will be easily deduced from any two triangles by substituting to one of them an homothetic triangle, the ratio being equal to the ratio of the distances of m to its orthopoles in the two given triangles, and the centre being the point where m meets the straight line joining the two orthopoles. 83. Some orthopole Theorems in special triangles(1). When

When are on a straight line;

EF passes through the mid-point of; passes through the mid-point of the are BC; is the mid-point of AN. When 2/ƒ¿=1/b+1/c, is on the symmedian from Am in AmBmCm; the orthopole of the circumdiameter perpendicular to IK lies on the straight line joining G to the point where AK meets the circumcircle. When 2ƒ¿2=b2+c2, is on the straight line joining A to the centre conic passing through the mid-points of the internalof the bisectrices and having its tangents at these points parallel to BC, CA, AB; lies on the symmedian from Am in AmBmCm and also on the straight line joining Ah to the mid-point of BmCm. When 3ƒ¿=b+c, the tangent at to the in-circle passes through Da; the orthopole of IO in IaIbIc lies on the circle having AI as a diameter. When ƒ¿(b+c)=b3+c2, is the image of K through BmCm;

passes through G; if K'a is the point where the tangent to the circumcircle at A meets BC, the circle having AK'a as a diameter touches IOg at; lies on the circle having as a diameter the distance between the mid-points of the two bisectrices of the angle A. When A=60•‹ or 120•‹, the straight line joining Am to the point divi-

ding GI in the ratio 1:3 passes through;

is the mid-point of AI; lies on AI. When 2AH=BH+CH, A,D are on a straight line. When cot2A=cot B•Ecot C, lies on AG. When ƒ¿R=(b+c)z, when ƒ¿R=(b+c)ƒÁa,

(1) For properties of special triangles, see Neuberg, Biblio Jraphie des triangle; spec:aux, SL, 1924; Goorma Khtigh, Proprietes nouuelles des triangles speciaux, SL, 1925. THE ORTHOPOLE. 125

When ƒ¡ is the mid-point of AD, lies on the straight line joining the image of D through I to the image of A through H.

INDEX.

(The numbers refer to sections).