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Advanced Euclidean Geometry Advanced Euclidean Geometry Paul Yiu Summer 2013 Department of Mathematics Florida Atlantic University A b c B a C 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.
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