Introduction to the Geometry of the Triangle

Introduction to the Geometry of the Triangle

147 Pages · 2011 · 983 KB · English

Introduction to the Geometry of the Triangle Paul Yiu Summer 2001 Department of Mathematics Florida Atlantic University Version 2.0402 April 2002

Introduction to the Geometry of the Triangle free download


Introduction to the Geometry of the Triangle Paul Yiu Summer 2001 Department of Mathematics Florida Atlantic University Version 2.0402 April 2002 Ta b l e o f C o n t e n t s Chapter 1The circumcircle and the incircle 1 1.1Preliminaries 1 1.2The circumcircle and the incircle of a triangle 4 1.3Euler’s formula and Steiner’s porism 9 1.4Appendix: Constructions with the centers of similitude of the circumcircle and the incircle 11 Chapter 2The Euler line and the nine-point circle 15 2.1The Euler line 15 2.2The nine-point circle 17 2.3Simson lines and reflections 20 2.4Appendix: Homothety 21 Chapter 3Homogeneous barycentric coordinates 25 3.1Barycentric coordinates with reference to a triangle 25 3.2Cevians and traces 29 3.3Isotomic conjugates 31 3.4Conway’s formula 32 3.5The Kiepert perspectors 34 Chapter 4Straight lines 43 4.1The equation of a line 43 4.2Infinite points and parallel lines 46 4.3Intersection of two lines 47 4.4Pedal triangle 51 4.5Perpendicular lines 54 4.6Appendix: Excentral triangle and centroid of pedal triangle 58 Chapter 5Circles I 61 5.1Isogonal conjugates 61 5.2The circumcircle as the isogonal conjugate of the line at infinity 62 5.3Simson lines 65 5.4Equation of the nine-point circle 67 5.5Equation of a general circle 68 5.6Appendix: Miquel theory 69 Chapter 6Circles II 73 6.1Equation of the incircle 73 6.2Intersection of incircle and nine-point circle 74 6.3The excircles 78 6.4The Brocard points 80 6.5Appendix: The circle triad (A(a),B(b),C(c)) 83 Chapter 7Circles III 87 7.1The distance formula 87 7.2Circle equation 88 7.3Radical circle of a triad of circles 90 7.4The Lucas circles 93 7.5Appendix: More triads of circles 94 Chapter 8Some Basic Constructions 97 8.1Barycentric product 97 8.2Harmonic associates 100 8.3Cevian quotient 102 8.4Brocardians 103 Chapter 9Circumconics 105 9.1Circumconic as isogonal transform of lines 105 9.2The infinite points of a circum-hyperbola 108 9.3The perspector and center of a circumconic 109 9.4Appendix: Ruler construction of tangent 112 Chapter 10General Conics 113 10.1Equation of conics 113 10.2Inscribed conics 115 10.3The adjoint of a matrix 116 10.4Conics parametrized by quadratic equations 117 10.5The matrix of a conic 118 10.6The dual conic 119 10.7The type, center and perspector of a conic 121 Chapter 11Some Special Conics 125 11.1Inscribed conic with prescribed foci 125 11.2Inscribed parabola 127 11.3Some special conics 129 11.4Envelopes 133 Chapter 12Some More Conics 137 12.1Conics associated with parallel intercepts 137 12.2Lines

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