Geometry Notes Paul Yiu Department of Mathematics Florida Atlantic University July 31, 2004 Contents 1 Triangle centers 1 1.1 December 1, 1999: X500 ........................... 1 1.2 May 20, 2000: Orthocentroidal circle . ................... 2 1.2.1 The Evans conic . .......................... 5 1.2.2 The type of the Evans conic . ................... 5 1.2.3 . ................................. 7 1.3 January 13, 2001: The bridge points . ................... 8 1.4 October 24, 2001 . .......................... 10 1.5 November 19, 2001: Tatiana Emelyanov on the Feuerbach point . 12 1.6 December 12, 2001: Edward Brisses’ construction of the Steiner point . 15 1.7 March 19, 2002: Taylor center . ................... 16 1.8 May 9, 2002: On X184 ............................ 16 1.9 ........................................ 16 1.10 May 11, 2002: Points on cubic . ................... 17 1.11 November 28, 2002 . .......................... 17 1.12 January 3, 2003: Bogomolny’s conjecture . ............ 18 1.13 January 22, 2003: Nikolaos Dergiades’ problem . ............ 19 1.14 February 26, 2003: Arnold’s Theorem ................... 20 1.15 March 5, 2003: Coordinates of the foci of Steiner in-ellipse ........ 21 1.16 March 12, 2003: Lev Emelyanov . ................... 21 1.16.1 March 14, 2003 . .......................... 22 1.17 March 13, 2003: David Wilson on X647 ................... 23 1.18 March 15, 2003: Malfatti-Rabinowitz points . ............ 23 1.19 June 8, 2003: ................................. 24 1.20 June 14, 2003: Steiner point and the nine-point center . ........ 24 1.21 September 29, 2003: APH and ED . ................... 24 1.22 October, 25, 2003: X1337 and X1338 ..................... 25 1.23 November 3, 2003: Eric Danneels . ................... 25 1.24 November 6, 2003 . .......................... 26 1.25 November 11, 2003: X1113 and X1822 .................... 26 1.26 December 1, 2003: JPE on two Lemoine centers . ............ 27 1.27 December 3, 2003: Darij Grinberg on generalization of orthocenter . 28 1.28 December 8, 2003: . .......................... 28 1.29 January 6, 2004: Eric Danneels . ................... 29 iv CONTENTS 1.30 January 20, 2004: Darij Grinberg’s Mittenpunkt note . ...... 29 1.31 February 11, 2004 . ........................ 29 1.32 February 14, 2004: Barycentric cube of Euler infinite point . ...... 29 1.33 February 26, 2004: Eric Danneels . ................. 30 1.34 March 19, 2004: Discrepancies with ETC ................. 30 1.35 March 21, 2004: X2028 and X2029 ...................... 31 1.36 March 22, 2004: Eric Daaneels . ................. 32 1.37 March 23, 2004: ............................... 33 1.37.1 JPE, Hyacinthos 6333, corrected in Hyacinthos 6337 . ...... 33 1.37.2 DG, Hyacinthos 7164 . ........................ 34 1.38 April 3, 2004: . ............................... 34 1.39 April 7, 2004: . ............................... 34 1.40 April 14, 2004: (Hyacinthos 9687) Symmedian point of the Apollonius triangle . ............................... 34 1.41 May 12, 2004: X57 = perspector of orthic triangle of intouch triangle . 35 1.42 May 28, 2004: ETC description of X552 and X553 ............. 35 1.43 June 2, 2004: Symmedian point . ................. 36 1.44 July 1, 2004: Edward Brisse on X(557) .................. 36 1.45 July 2, 2004: Eric Danneels . ........................ 36 1.46 July 16, 2004: Stevanovic . ........................ 37 1.47 July 17, 2004: X1157 ............................. 39 1.48 July 17, 2004: The deLongchamps point . ................. 39 1.49 July 21, 2004: APH on eight circumcenters ................. 40 1.50 July 24, 2004: Circumcenters tangential and Kosnita triangle . ...... 41 1.51 July 24, 2004: . ............................... 41 2 Homothetic triangles 43 2.1 March 11, 2000: Two Pairs of Poristic Conics Associated with parallel intercepts . ............................... 43 2.1.1 The conic CP ............................. 43 2.1.2 The center of the conic CP ...................... 44 2.1.3 The conic inscribed in the hexagon W (P ) .............. 45 2.1.4 Another hexagon AC BACB .................... 46 2.1.5 Examples ............................... 47 2.2 March 18, 2000: A quadruple of homothetic triangles with four collinear centers of similitude . ........................ 48 2.3 March 21, 2000: Homothetic Centers . ................. 51 2.3.1 The Euler and Carnot triangles . ................. 52 2.4 April 8, 2004: . ............................... 53 2.5 July 24, 2004: Gob homothetic center . ................. 54 3 Parallelians 55 3.1 March 9, 2000: Lemoine’s Problem . ................. 55 3.2 January 18, 2002: Parallelian Hyacinthos 4667 . ............. 57 3.3 June 19, 2002: Conic through the endpoints of parallelians through P ... 59 CONTENTS v 3.4 September 6, 2002: Parallelians . ................... 60 4 Pedal triangles 61 4.1 January 5, 2001: Perspectivity of pedal triangle and anticevian triangle . 61 4.2 August 31, 2002: Pedal residuals . ................... 61 4.3 June 2, 2004: Pedal triangle . ................... 61 4.3.1 A variation . .......................... 62 5 Operations 63 5.1 October 16, 1999: APH’s product . ................... 63 5.2 April 5, 2000: Harmon of desmic systems . ............ 66 5.3 October 24, 2001 . .......................... 68 5.4 October 27, 2001 . .......................... 70 5.5 February 18, 2002: Isoscelizers . ................... 71 5.6 October 26, 2002: Orthogonal quadrisection of a triangle . ........ 72 5.7 January 22, 2003: Euler’s quadrisection problem . ............ 72 5.8 January 24, 2003: Darij Grinberg’s isotome complement . ........ 73 5.9 February 13, 2003: Myakishev’s cevasix note . ............ 74 5.10 March 6, 2003: Isoscelizers . ................... 74 5.11 March 9, 2003: Left and right isoscelizers . ............ 78 5.12 October 24, 2003: Bernard’s AOC paper . ............ 79 5.13 November 4, 2003: . .......................... 81 5.13.1 . ................................. 81 5.14 November 8, 2003: The barycentric cube of an infinite point ........ 82 5.15 November 14, 2003: APH’s symmetrologic triangles ............ 82 5.16 November 15, 2003: Euler quartic . ................... 83 5.17 November 19, 2003: Paper from Turkey . ............ 84 5.18 November 23, 2003: Intersections of circles and lines . ........ 87 5.19 December 4, 2003: Least square line . ................... 88 5.20 February 6, 2004: APH circumcevian triangle problem . ........ 88 5.21 April 5, 2004: An interesting distance formula . ............ 89 5.22 April 19, Abu-Saymeh and Hajji . ................... 89 5.23 April 28, 2004: Isogonal conjugate of the inferior . ............ 89 5.24 June 2, 2004: JPE’s Hyacinthos 10000 ................... 89 5.25 June 10, 2004: Michel Garitte . ................... 90 5.26 July 29, 2004: APH’s second and third orthocorrespondences . 90 6 Euler lines 93 6.1 January 22, 2004: . .......................... 93 6.2 January 28, 2004: Final touches on the concurrent Euler lines paper . 94 6.3 January 30, 2004 . .......................... 97 6.4 July 25, 2004: Euler line . .......................... 97 vi CONTENTS 7 Residuals 99 7.1 November 20, 2002: Mario Dalc´in..................... 99 7.2 December 3, 2002: Mario Dalcin . ................. 99 8 Orthology 101 8.1 May 20, 2004: Orthology . ........................101 8.2 May 26, 2004: orthology . ........................102 8.3 June 4, 2004: APH, Hyacinthos 9858 . .................102 8.3.1 A variation . ........................102 8.4 July 11, 2004: . ...............................102 8.5 July 27, 2004: JPE on orthologic triangles .................102 8.6 July 30, 2004: JPE on orthology centers . .................104 9 Transformation of coordinates 105 9.1 May 17, 2002: Transformation of coordinates . .............105 9.1.1 McCay cubic with reference to the excentral triangle . ......105 10 Cevian nests 107 10.1 February 28, 2002: FvL, Hyacinthos 4839 .................107 10.2 September 15, 2003: The cevian nest theorem . .............108 10.3 March 18 2004: Cevian quotients . .................109 10.3.1 G/P .................................109 10.3.2 H/P .................................109 10.4 March 26, 2004: ...............................110 11 Reflections 113 11.1 October 19, 1999: Musselman’s theorem . .................113 11.2 December 9, 2000: Reflections . .................115 11.3 September 28, 2001: Reflection triangle . .................121 11.4 Gilles Bouttes, Hyacinthos 3997 (9/28/01) .................123 11.5 October 26, 2001: Reflection triangles . .................123 11.6 June 1, 2002: Scimemi’s paper . .................125 11.7 August 9, 2002. Paper folding the roots of a cubic equation . ......127 11.8 January 18, 2003: Musselman’s 1940 Monthly paper . ......129 11.9 March 1, 2003: Reflections . ........................130 11.10 September 24, 2003: JPE on reflections of a point in its cevian triangle . 131 11.11 December 30, 2003: Hatzipolakis reflection point X1986 ..........132 11.12 December 31, 2003: Reflections of circumcircles of pedal residuals . 133 11.13 January 8, 2004: D. Moody Bailey . .................135 11.14 January 31, 2004: . ........................135 11.15 February 9, 2004: . ........................136 11.16 February 17, 2004: Reflections in altitudes .................136 11.17 February 17, 2004: . ........................137 11.18 March 18, 2004: Reflection conjugates . .................138 11.19 March 26, 2004: Review on Simson and reflection lines . ......138 CONTENTS vii 11.19.1 Simson lines of antipodal points ...................139