Introduction to the Geometry of the Triangle Paul Yiu Summer 2004 Department of Mathematics Florida Atlantic University Version 4.0510 May 2004 Contents 1 The Circumcircle and the Incircle 1 1.1 Preliminaries ................................. 1 1.1.1 Coordinatization of points on a line . ............ 1 1.1.2 Centers of similitude of two circles . ............ 2 1.1.3 Tangent circles . .......................... 2 1.1.4 Harmonic division .......................... 2 1.1.5 Homothety . .......................... 4 1.1.6 The power of a point with respect to a circle . ............ 4 1.2 Menelaus and Ceva theorems . ................... 6 1.2.1 Menelaus and Ceva Theorems . ................... 6 1.2.2 Desargues Theorem . ................... 7 1.3 The circumcircle and the incircle of a triangle . ............ 8 1.3.1 The circumcircle . .......................... 8 1.3.2 The incircle and the Gergonne point . ............ 8 1.3.3 The Heron formula .......................... 10 1.3.4 The excircles and the Nagel point . ............ 12 1.4 The medial and antimedial triangles . ................... 14 1.4.1 The medial triangle, the nine-point center, and the Spieker point . 14 1.4.2 The antimedial triangle and the orthocenter . ............ 14 1.4.3 The Euler line . .......................... 15 1.5 The nine-point circle . .......................... 17 1.5.1 The Euler triangle as a midway triangle . ............ 17 1.5.2 The orthic triangle as a pedal triangle . ............ 17 1.5.3 The nine-point circle . ................... 17 1.5.4 The orthic triangle .......................... 19 1.6 The OI-line ................................. 20 1.6.1 The centers of similitude of the circumcircle and the incircle . 20 1.6.2 Reflection of I in O ......................... 20 1.6.3 Orthocenter of intouch triangle . ................... 21 1.6.4 Centroid of excentral triangle . ................... 21 1.6.5 Homothetic center of excentral and intouch triangle . ........ 22 2 Homogeneous Barycentric Coordinates 27 2.1 Barycentric coordinates with reference to a triangle ............ 27 2.1.1 Homogeneous barycentric coordinates . ............ 27 iv CONTENTS 2.1.2 The area formula . ........................ 31 2.2 Equations of Straight lines . ........................ 32 2.2.1 Two-point form . ........................ 32 2.2.2 Intercept form: trilinear pole and polar . ............. 32 2.2.3 Intersection of two lines . ................. 33 2.3 Cevian and anticevian triangles . ................. 35 2.3.1 Cevian triangle . ........................ 35 2.3.2 Anticevian triangle . ........................ 36 2.3.3 Construction of anticevian triangle . ................. 37 2.4 Perspectivity . ............................... 39 2.4.1 Triangles bounded by lines parallel to the sidelines . ...... 39 2.4.2 Perspector and perspectrix . ................. 40 2.4.3 Cevian nest theorem . ........................ 40 2.5 Conway’s formula . ........................ 42 2.5.1 Conways’s Notation . ........................ 42 2.5.2 Jacobi’s Theorem . ........................ 44 2.5.3 Isosceles triangles erected on the sides and Kiepert perspectors . 44 3 Some Basic Constructions 49 3.1 Isotomic conjugates . ........................ 49 3.1.1 The isotomic conjugate of the orthocenter . ............. 49 3.1.2 Equal-parallelians point . ................. 50 3.1.3 Appendix: Yff’s analogue of the Brocard points . ...... 52 3.2 Reflections and isogonal conjugates . ................. 53 3.2.1 Examples of isogonal conjugates . ................. 53 3.3 Examples of isogonal conjugates . ................. 56 3.3.1 Isogonal conjugates of the Gergonne and Nagel points . ...... 56 3.3.2 The Brocard points . ........................ 57 3.3.3 Isogonal conjugates of the Kiepert perspectors . ...... 58 3.4 Barycentric product . ........................ 59 3.4.1 Examples ............................... 59 3.4.2 Barycentric square root ........................ 60 3.5 Cevian quotient ............................... 63 3.5.1 Examples ............................... 63 3.6 The Brocardians ............................... 64 4 Straight Lines 65 4.1 Parametrization of a line . ........................ 65 4.2 Infinite points and parallel lines . ................. 65 4.2.1 The infinite point of a line . ................. 65 4.2.2 Parallel lines . ........................ 66 4.3 Perpendicular lines . ........................ 68 4.3.1 The Orthopole of a line ........................ 69 4.3.2 2 . ............................... 69 4.4 The distance formula . ........................ 71 CONTENTS v 4.5 The pedal and reflection of a point on a line . ............ 73 4.6 Pedal triangles . .......................... 74 4.6.1 Pedal triangle . .......................... 74 4.6.2 Examples . .......................... 74 4.6.3 Pedal residuals . .......................... 75 4.6.4 Centroid of pedal triangle . ................... 76 4.7 Reflection triangles . .......................... 78 4.8 Antipedal triangles . .......................... 79 4.9 Appendices ................................. 80 4.9.1 The excentral triangle . ................... 80 4.9.2 Perspectors associated with inscribed squares ............ 80 5 Circles 83 5.1 Isogonal conjugates of infinite points and the circumcircle . ........ 83 5.1.1 Equation of the circumcircle . ................... 83 5.1.2 A parametrization of the circumcircle . ............ 83 5.1.3 Antipodal points on the circumcircle . ............ 84 5.2 Circumcevian triangle . .......................... 85 5.3 Simson lines ................................. 86 5.3.1 Simson lines of antipodal points ................... 87 5.4 Equation of circles . .......................... 88 5.4.1 Equation of a general circle . ................... 88 5.4.2 Equation of the nine-point circle................... 88 5.4.3 The power of a point with respect to a circle . ............ 89 5.4.4 The incircle and the excircles . ................... 89 5.5 The Feuerbach theorem . .......................... 91 5.5.1 Intersection of the incircle and the nine-point circle . ........ 91 5.5.2 Condition for tangency of a line and the incircle . ........ 91 6 Special Circles 93 6.1 General circle equations . .......................... 93 6.1.1 Circle with given center and radius . ............ 93 6.1.2 Circle with a given diameter . ................... 93 6.1.3 The power of a point with respect to a circle . ............ 94 6.2 The Taylor circle . .......................... 95 6.3 The Dou circle . .......................... 96 6.3.1 August 17, 2002: Edward Brisse ................... 97 6.4 The Brocard points . .......................... 98 7 Triads of Circles 101 7.1 The excircles .................................101 7.2 Appendix: Miquel Theory ..........................103 7.2.1 Miquel Theorem . ..........................103 7.2.2 Miquel associate . ..........................103 7.2.3 Cevian circumcircle . ...................104 vi CONTENTS 7.2.4 Cyclocevian conjugate ........................104 7.3 Appendix: The circle triad (A(a),B(b),C(c)) ...............107 7.3.1 The Steiner point . ........................108 7.4 Radical circle of a triad of circles . .................109 7.4.1 Radical center . ........................109 7.4.2 Radical circle . ........................109 7.4.3 The excircles . ........................110 7.4.4 The de Longchamps circle . .................110 7.5 2 .......................................112 7.6 Appendix: More triads of circles . .................113 7.6.1 The triad {A(AH )} ..........................113 7.7 The triad of circles (AG(AH ),BG(BH ),CG(CH )) .............116 9 Circumconics 117 9.1 Circumconics as isogonal transforms of lines . .............117 9.2 The perspector and center of a circumconic .................120 9.2.1 Examples ...............................120 9.3 The infinite points and asymptotes of a circum-hyperbola . ......122 9.3.1 Stother’s Theorem . ........................122 10 General Conics 123 10.1 Equation of conics . ........................123 10.1.1 Conic through the traces of P and Q .................123 10.2 Inscribed conics ...............................125 10.2.1 The Steiner in-ellipse . ........................125 10.3 The matrix of a conic . ........................126 10.3.1 Line coordinates . ........................126 10.3.2 The matrix of a conic . ........................126 10.3.3 Tangent at a point . ........................126 10.4 The dual conic . ...............................127 10.4.1 Pole and polar . ........................127 10.4.2 Condition for a line to be tangent to a conic .............127 10.4.3 The dual conic . ........................127 10.4.4 The dual conic of a circumconic . .................127 10.4.5 Appendix: The adjoint of a matrix . .................128 10.5 The type, center and perspector of a conic .................130 10.5.1 The type of a conic . ........................130 10.5.2 The center of a conic . ........................130 10.5.3 The perspector of a conic . .................130 10.6 Conics parametrized by quadratic functions .................132 10.6.1 Locus of Kiepert perspectors . .................132 10.6.2 . ...............................133 CONTENTS vii 11 Some Special Conics 135 11.1 Inscribed conic with prescribed foci . ...................135 11.1.1 Theorem . ..........................135 11.1.2 The Brocard ellipse ..........................135 11.1.3 2 . .................................136 11.1.4 The Lemoine ellipse . ...................136 11.1.5 The inscribed conic with center N ..................137