Advanced Euclidean Geometry

Paul Yiu

Summer 2013

Department of Mathematics Florida Atlantic University

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Version 13.0611

Summer 2013 Contents

1 Some Basic Theorems 101 1.1 ThePythagoreanTheorem...... 101 1.2 Constructions of geometric mean ...... 104 1.3 Thegoldenratio ...... 106 1.3.1 The regular pentagon ...... 106 1.4 Basic construction principles ...... 108 1.4.1 Perpendicular bisector locus ...... 108 1.4.2 Angle bisector locus ...... 109 1.4.3 Tangencyofcircles...... 110 1.4.4 Construction of tangents of a circle ...... 110 1.5 The intersecting chords theorem ...... 112 1.6 Ptolemy’stheorem...... 114

2 The laws of sines and cosines 115 2.1 Thelawofsines ...... 115 2.2 Theorthocenter ...... 116 2.3 Thelawofcosines...... 117 2.4 Thecentroid ...... 120 2.5 Theanglebisectortheorem ...... 121 2.5.1 The lengths of the bisectors ...... 121 2.6 ThecircleofApollonius...... 123

3 The tritangent circles 125 3.1 Theincircle ...... 125 3.2 Euler’sformula ...... 128 3.3 Steiner’sporism ...... 129 3.4 Theexcircles...... 130 3.5 Heron’s formula for the area of a triangle ...... 131

4 Thearbelos 133 4.1 Archimedes’twincircles ...... 133 1 1 1 4.1.1 Harmonic mean and the equation a + b = t ...... 134 4.1.2 Construction of the Archimedean twin circles ...... 134 4.2 Theincircle ...... 135 4.2.1 Construction of the incircle of the arbelos ...... 136 iv CONTENTS

4.2.2 Alternative constructions of the incircle ...... 138 4.3 Archimedeancircles...... 140

5 Menelaus’ theorem 201 5.1 Menelaus’theorem ...... 201 5.2 Centers of similitude of two circles ...... 203 5.2.1 Desargue’stheorem...... 203 5.3 Ceva’stheorem...... 204 5.4 Sometrianglecenters ...... 205 5.4.1 Thecentroid ...... 205 5.4.2 Theincenter ...... 205 5.4.3 The Gergonne point ...... 206 5.4.4 TheNagelpoint...... 207 5.5 Isotomicconjugates ...... 208

6 The Euler line and the nine-point circle 211 6.1 TheEulerline ...... 211 6.1.1 Inferior and superior triangles ...... 211 6.1.2 The orthocenter and the Euler line ...... 212 6.2 Thenine-pointcircle...... 213 6.3 Distances between triangle centers ...... 215 6.3.1 Distance between the circumcenter and orthocenter ...... 215 6.3.2 Distance between circumcenter and tritangent centers...... 216 6.3.3 Distance between orthocenter and tritangent centers ...... 217 6.4 Feuerbach’stheorem...... 219

7 Isogonal conjugates 221 7.1 Directedangles ...... 221 7.2 Isogonalconjugates ...... 222 7.3 The symmedian point and the centroid ...... 223 7.4 Isogonal conjugates of the Gergonne and Nagel points ...... 225 7.4.1 The Gergonne point and the insimilicenter T+ ...... 225 7.4.2 The Nagel point and the exsimilicenter T ...... 227 7.5 TheBrocardpoints ...... 228−

8 The excentral and intouch triangles 231 8.1 Theexcentraltriangle ...... 231 8.1.1 The circumcenter of the excentral triangle ...... 231 8.1.2 Relation between circumradius and radii of tritangent circles . . . . 233 8.2 The homothetic center of the intouch and excentral triangles ...... 234

9 Homogeneous Barycentric Coordinates 301 9.1 Absolute and homogeneous barycentric coordinates ...... 301 9.1.1 Thecentroid ...... 301 9.1.2 Theincenter ...... 302 CONTENTS v

9.1.3 The barycenter of the perimeter ...... 303 9.1.4 The Gergonne point ...... 304 9.2 Ceviantriangle...... 305 9.2.1 Thecircumcenter...... 306 9.2.2 The Nagel point and the extouch triangle ...... 307 9.2.3 The orthocenter and the orthic triangle ...... 308 9.3 Homotheties ...... 309 9.3.1 Superiors and inferiors ...... 309 9.3.2 Triangles bounded by lines parallel to the sidelines ...... 312

10 Some applications of barycentric coordinates 315 10.1 Construction of mixtilinear incircles ...... 315 10.1.1 The insimilicenter and the exsimilicenter of the circumcircle and incircle ...... 315 10.1.2 Mixtilinear incircles ...... 316 10.2 Isotomic and isogonal conjugates ...... 317 10.3 Isotomicconjugates ...... 317 10.4 Equal-parallelians point ...... 319

11 321 11.1 TheFeuerbachpoint...... 321 11.2 The OI line ...... 322 11.2.1 The circumcenter of the excentral triangle ...... 322 11.2.2 The centers of similitude of the circumcircle and the incircle . . . . 322 11.2.3 The homothetic center T of excentral and intouch triangles . . . . . 322

12 Some interesting circles 323 12.1 A fundamental principle on 6 concyclic points ...... 323 12.1.1 The radical axis of two circles ...... 323 12.1.2 Test for 6 concyclic points ...... 324 12.2 TheTaylorcircle...... 325 12.3 TwoLemoinecircles...... 326 12.3.1 ThefirstLemoinecircle ...... 326 12.3.2 ThesecondLemoinecircle...... 327 12.3.3 Construction of K ...... 327 12.3.4 The center of the first Lemoine circle ...... 328

13 Straight line equations 401 13.1 Area and barycentric coordinates ...... 401 13.2 Equations of straight lines ...... 402 13.2.1 Two-point form ...... 402 13.2.2 Intersection of two lines ...... 403 13.3 Perspectivetriangles...... 404 13.3.1 The Conway configuration ...... 405 13.4 Perspectivity...... 407 vi CONTENTS

13.4.1 The Schiffler point: intersection of four Euler lines ...... 408

14 Cevian nest theorem 409 14.1 Trilinearpoleandpolar ...... 409 14.1.1 Trilinear polar of a point ...... 409 14.1.2 Tripole of a line ...... 411 14.2 Anticeviantriangles ...... 412 14.2.1 Construction of anticevian triangle ...... 412 14.3 Cevianquotients...... 415 14.3.1 Theceviannesttheorem ...... 415 14.3.2 Basic properties of cevian quotients ...... 419