SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT
QUANG HUNG TRAN
Abstract. We establish a relationship between the two important central lines of the triangle, the Euler line and the Brocard axis, in a configuration with an arbitrary rectangle and a random point. The classical Cartesian coordinate system method shows its strength in these theorems. Along with that, some related problems on rectangles and a random point are proposed with similar solutions using Cartesian coordinate system.
1. Introduction In the geometry of triangles, the Euler line [8] plays an important role and is almost the most classic concept in this field. The Euler line passes through the centroid, orthocenter, and circumcenter of the triangle. The symmedian point [9] of a triangle is the concurrent point of its symmedian lines, the Brocard axis [10] is the line connecting the circumcircle and the symmedian point of that triangle. The Brocard axis is also a central line that plays an important role in the geometry of triangle. The coordinate method was invented by Ren´eDescartes from the 17th century [1,2], up to now, the classical coordinate method of Descartes is still one of the most important method of mathematics and geometry. In this paper, we apply Cartesian coordinate system to solve an interesting the- orem for an arbitrary rectangle and a random point in which two important central lines in the triangle are mentioned as follows. The idea of using the Cartesian coor- dinate system is also used in a similar way to solve the new problems we introduced in Section 3.
2. Main theorem and proof Theorem 1. Let ABCD be a rectangle with center I. Let P be a random point in its plane. arXiv:2106.10810v1 [math.HO] 21 Jun 2021 1) Euler line of triangles P AB and PCD meet at Q. Euler line of triangles PBC and P AD meet at R. Then, line QR goes through center I. 2) Brocard axis of triangles P AB and PCD meet at M. Brocard axis of triangles PBC and P AD meet at N. Then, line MN goes through P . Proof. We will use Cartesian coordinate for the solutions. Since ABCD is an arbitrary rectangle and P is a random point, we can take P (0, 0), A(a, b), B(c, b), C(c, d), and D(a, d) for all real numbers a, b, c, and d.
Date: June 22, 2021. 2010 Mathematics Subject Classification. 51M04, 51-03. 1 2 QUANG HUNG TRAN
We get perpendicular bisectors da, db, dc, and dd of PA, PB, PC, and PD, respectively, are a2 + b2 a (2.1) d : y = − · x, a 2b b b2 + c2 c (2.2) d : y = − · x, b 2b b c2 + d2 c (2.3) d : y = − · x, c 2d d a2 + d2 a (2.4) d : y = − · x. d 2d d Thus circumcenters Oa, Ob, Oc, and Od of triangles P AB, PBC, PCD, and PDA, respectively, are a + c −ac + b2 (2.5) O = d ∩ d = , , a a b 2 2b −bd + c2 b + d (2.6) O = d ∩ d = , , b b c 2c 2 a + c −ac + d2 (2.7) O = d ∩ d = , , c c d 2 2d −bd + a2 b + d (2.8) O = d ∩ d = , . d d a 2a 2 1) (See Figure1). We have that centroid Ga, Gb, Gc, and Gd of triangles P AB, PBC, PCD, and PDA, respectively, are a + c 2b (2.9) G = , , a 3 3 2c b + d (2.10) G = , , b 3 3 a + c 2d (2.11) G = , , c 3 3 2a b + d (2.12) G = , . d 3 3 Therfore, Euler lines a, b, c, and d of triangles P AB, PBC, PCD, and PDA, respectively, are b2 + ac b2 + 3ac (2.13) : y = − · x, a b ab + bc bc2 + bd2 + b2d + c2d bc + cd (2.14) : y = − · x, b c2 + 3bd c2 + 3bd d2 + ac d2 + 3ac (2.15) : y = − · x, c d ad + cd SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 3
y
Q
D(a, d) C(c, d) b
a+c b+d I 2 , 2 d
A(a, b) B(c, b)
c x P (0, 0) a R
Figure 1. Rectangle with Euler lines
bd2 + a2b + a2d + b2d ab + ad (2.16) : y = − · x. d a2 + 3bd a2 + 3bd We get the intersections a2c − abd + ac2 − bcd 2abc + 2acd (2.17) Q = ∩ = , , a c 3ac − bd 3ac − bd −2abd − 2bcd abc + acd − b2d − bd2 (2.18) R = ∩ = , , b d ac − 3bd ac − 3bd and then the line connecting points Q and R is b + d (2.19) QR : y = − x + b + d. a + c a+c b+d Now it not hard to see that line PQ goes through center I 2 , 2 . 2) (See Figure2). Using the barycentric coordinates of symmedian point in [9], we have that the coordinates of symmedian points Sa, Sb, Sc, and Sd of triangles P AB, PBC, PCD, and PDA, respectively, are (2.20) PA2 · B + PB2 · A + AB2 · P a2c + ab2 + ac2 + b2c a2b + 2b3 + bc2 S = = , , a PA2 + PB2 + AB2 2a2 − 2ac + 2b2 + 2c2 2a2 − 2ac + 2b2 + 2c2
15 4 QUANG HUNG TRAN
y
N D(a, d) C(c, d)
βb
βd A(a, b) B(c, b)
P (0, 0) x
βc βa
M
Figure 2. Rectangle with Brocard axis
(2.21) PB2 · C + PC2 · B + BC2 · P b2c + 2c3 + cd2 b2d + bc2 + bd2 + c2d S = = , , b PB2 + PC2 + BC2 2b2 − 2bd + 2c2 + 2d2 2b2 − 2bd + 2c2 + 2d2 (2.22) PC2 · D + PD2 · C + CD2 · P a2c + ac2 + ad2 + cd2 a2d + c2d + 2d3 S = = , , c PC2 + PD2 + CD2 2a2 − 2ac + 2c2 + 2d2 2a2 − 2ac + 2c2 + 2d2 (2.23) PD2 · A + PA2 · D + DA2 · P 2a3 + ab2 + ad2 a2b + a2d + b2d + bd2 S = = , . d PD2 + PA2 + DA2 2a2 + 2b2 − 2bd + 2d2 2a2 + 2b2 − 2bd + 2d2 Therfore, Brocard axis βa, βb, βc, and βd of triangles P AB, PBC, PCD, and PDA, respectively, are b4 + a2b2 + a2c2 + b2c2 −b4 − ac3 + a2c2 − a3c − 2ab2c (2.24) β : y = + · x, a 2bc2 + 2a2b − 4abc bc3 + a3b − abc2 − a2bc (2.25) bc4 + b2d3 + b3c2 + b3d2+ +c2d3 + c4d + bc2d2 + b2c2d −cd3 − b3c + bcd2 + b2cd β : y = + · x, b 2c4 + 2bd3 − 2b2d2 + 2b3d + 4bc2d c4 + bd3 − b2d2 + b3d + 2bc2d SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 5
d4 + a2c2 + a2d2 + c2d2 −d4 − ac3 + a2c2 − a3c − 2acd2 (2.26) β : y = + · x, c 2a2d + 2c2d − 4acd a3d + c3d − ac2d − a2cd (2.27) a2b3 + a2d3 + a4b + a4d+ +b2d3 + b3d2 + a2bd2 + a2b2d −ab3 − ad3 + abd2 + ab2d β : y = + · x. d 2a4 + 2bd3 − 2b2d2 + 2b3d + 4a2bd a4 + bd3 − b2d2 + b3d + 2a2bd We get the intersections (2.28) −a3bd + a3c2 − a2bcd + a2c3− a3bc + a3cd + a2bc2 + a2c2d+ 3 2 2 2 3 3 2 3 2 −ab d − ab d − abc d − abd − +ab c + ab cd + abc + abcd + 3 2 2 3 3 3 3 3 2 2 3 −b cd − b cd − bc d − bcd +ac d + acd − b d − b d M = βa∩βc = , , 2a3c − 2a2c2 − 4abcd+ 2a3c − 2a2c2 − 4abcd+ +2ac3 − 2b3d − 2b2d2 − 2bd3 +2ac3 − 2b3d − 2b2d2 − 2bd3 (2.29) −a3bd + a3c2 − a2bcd + a2c3− a3bc + a3cd + a2bc2 + a2c2d+ 3 2 2 2 3 3 2 3 2 −ab d − ab d − abc d − abd − +ab c + ab cd + abc + abcd + 3 2 2 3 3 3 3 3 2 2 3 −b cd − b cd − bc d − bcd +ac d + acd − b d − b d N = βb∩βd = , , 2a3c + 2a2c2 + 4abcd+ 2a3c + 2a2c2 + 4abcd+ +2ac3 − 2b3d + 2b2d2 − 2bd3 +2ac3 − 2b3d + 2b2d2 − 2bd3 and then the line connecting points M and N is b2d3 + b3d2 − abc3 − acd3 − ab3c − ac3d− −a2bc2 − a2c2d − a3bc − a3cd − abcd2 − ab2cd (2.30) MN : y = · x. −a2c3 − a3c2 + abd3 + ab2d2 + ab3d + bcd3+ +bc3d + a3bd + b2cd2 + b3cd + abc2d + a2bcd It is easy to see that MN goes through P (0, 0). This completes the proof. 6 QUANG HUNG TRAN
3. Some others theorems on rectangle and a random point We introduce some more theorems on rectangle and a random point, all of them can be solved by Cartesian coordinate as we work above.
Theorem 2 (Generalization of Theorem1) . Let A1B1C1D1 and A2B2C2D2 be two rectangles with the same center I. Let P be a random point in its plane. Euler lines of triangles PA1B1 and PC1D1 meet at Q. Euler lines of triangles PA2D2 and PB2C2 meet at R. Then, three points Q, R, and I are collinear (See Figure 3).
B1 A1
A2 Q B2 P
I
R D2 C2
D1 C1
Figure 3. SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 7
Theorem 3. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Ha, Hb, Hc, and Hd be the orthocenters of triangles P AB, PBC, PCD, and PDA, respectively. Let Q and R be the midpoints of HaHc and HbHd, respectively. Then, three points Q, R, and I are collinear (See Figure4).
Hd
B C
R
I H Hc a P Q
Hb
A D
Figure 4. 8 QUANG HUNG TRAN
Theorem 4. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Oa, Ob, Oc, and Od be the circumcenters of triangles P AB, PBC, PCD, and PDA, respectively. Let Q and R be the midpoints of OaOc and ObOd, respectively. Let S be the midpoint of QR. Circumcircles of triangles POaOc and PObOd meets again at T . Then, three points S, T , and I are collinear (See Figure 5).
Figure 5. SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 9
Theorem 5. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Na, Nb, Nc, and Nd be the nine-point centers of triangles P AB, PBC, PCD, and PDA, respectively. Let M and N be the midpoints of NaNc and NbNd, respectively. Let Q be intersection of perpendicular bisectors of NaNc and N N . Then, lines MN and IQ are parallel (See Figure6). MakeUpb d
Hide
Na
D C M Q
Nc
I
Nd Nb N
P
A B
Figure 6. 10 QUANG HUNG TRAN
Theorem 6. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Pa, Pb, Pc, and Pd be the isogonal conjugate of I with respect to triangles P AB, PBC, PCD, and PDA, respectively. Then, line IP bisects the segments PaPc and PbPd (See Figure7).
Pc
D C
Pd I N Pb P M
A B
Pa
Figure 7. SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 11
Theorem 7. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Pa, Pb, Pc, and Pd be the reflections of P in the lines AB, BC, CD, and DA, respectively. Euler line of triangles PaAB and PcCD meet at Q. Euler line of triangles PbBC and PdDA meet at R. Then, line IP bisects the segment QR (See Figure8).
Pc
R D C J
Q
P I Pd Pb
A B
Pa
Figure 8. 12 QUANG HUNG TRAN
Theorem 8. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Pa, Pb, Pc, and Pd be the reflections of P in the midpoints of sides AB, BC, CD, and DA, respectively. Euler line of triangles PaAB and PcCD meet at Q. Euler line of triangles PbBC and PdDA meet at R. Then, reflection of P in I lies on line QR (See Figure9).
Pc
D C
Q Pd P' Pb I P
A B R
Pa
Figure 9. SOME PROPERTIES OF RECTANGLE AND A RANDOM POINT 13
Theorem 9. Let ABCD be a rectangle with center I. Let P be a random point in its plane. Let Pa, Pb, Pc, Pd, Pac, and Pbd be the reflections of P in the lines AB, BC, CD, DA, AC, and BD, respectively. Orthogonal projections of Pac on sides of quadrilateral PaPbPcPd form a quadrilateral which has two diagonals meet at Q. Orthogonal projections of Pbd on sides of quadrilateral PaPbPcPd form a quadrilateral which has two diagonals meet at R (See Figure 10). Then,
i) Two points Pac and Pbd are isogonal conjugate with respect to quadrilateral PaPbPcPd. ii) There point P , Q, and R are collinear. iii) Two lines QPac and RPbd are parallel.
Pc
C
D Pbd R
Pd P I Pb
Pac Q
A B
Pa
Figure 10. 14 QUANG HUNG TRAN
References [1] Ren´eDescartes, Discourse de la M´ethode (Leiden, Netherlands): Jan Maire, 1637, appended book: La G´eom´etrie,book one, p. 299. [2] Sorell, T.: Descartes: A Very Short Introduction (2000). New York: Oxford University Press. p. 19. [3] Coxeter, H. S. M. and Greitzer, S. L.: Geometry Revisited, Washington, DC: Math. Assoc. Amer., 1967., p. 53. [4] Johnson, R. A.: Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle, Boston, MA: Houghton Mifflin, 1929., p. 172. [5] Coxeter, H. S. M.: Projective geometry, Blaisdell, New York, 1964., p. 78. [6] Coxeter, H. S. M.: Non-Euclidean Geometry, University of Toronto Press, 1942., p. 29. [7] Weisstein, E. W.: Rectangle, from MathWorld–A Wolfram Web Resource, https:// mathworld.wolfram.com/Rectangle.html. [8] Weisstein, E. W.: Euler Line, from MathWorld–A Wolfram Web Resource, https:// mathworld.wolfram.com/EulerLine.html. [9] Weisstein, E. W.: Symmedian Point, from MathWorld–A Wolfram Web Resource, https: //mathworld.wolfram.com/SymmedianPoint.html. [10] Weisstein, E. W.: Brocard Axis, from MathWorld–A Wolfram Web Resource, https:// mathworld.wolfram.com/BrocardAxis.html. [11] Q. H. Tran and L. Gonz´alez, Two generalizations of the Butterfly Theorem, arXiv:2012.08365 [math.HO]. [12] Geogebra Geometry, https://www.geogebra.org/geometry. [13] Sage Notebook v6.10, http://sagemath.org.
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