Geometry of the Triangle Paul Yiu 2016 Department of Mathematics Florida Atlantic University A b c B a C August 18, 2016 Contents 1 Some basic notions and fundamental theorems 101 1.1 Menelaus and Ceva theorems ........................101 1.2 Harmonic conjugates . ........................103 1.3 Directed angles ...............................103 1.4 The power of a point with respect to a circle ................106 1.4.1 Inversion formulas . ........................107 1.5 The 6 concyclic points theorem . ....................108 2 Barycentric coordinates 113 2.1 Barycentric coordinates on a line . ....................113 2.1.1 Absolute barycentric coordinates with reference to a segment . 113 2.1.2 The circle of Apollonius . ....................114 2.1.3 The centers of similitude of two circles . ............115 2.2 Absolute barycentric coordinates . ....................120 2.2.1 Homotheties . ........................122 2.2.2 Superior and inferior . ........................122 2.3 Homogeneous barycentric coordinates . ................122 2.3.1 Euler line and the nine-point circle . ................124 2.4 Barycentric coordinates as areal coordinates ................126 2.4.1 The circumcenter O ..........................126 2.4.2 The incenter and excenters . ....................127 2.5 The area formula ...............................128 2.5.1 Conway’s notation . ........................129 2.5.2 Conway’s formula . ........................130 2.6 Triangles bounded by lines parallel to the sidelines . ............131 2.6.1 The symmedian point . ........................132 2.6.2 The first Lemoine circle . ....................133 3 Straight lines 135 3.1 The two-point form . ........................135 3.1.1 Cevian and anticevian triangles of a point . ............136 3.2 Infinite points and parallel lines . ....................138 3.2.1 The infinite point of a line . ....................138 3.2.2 Infinite point as vector ........................141 3.3 Perpendicular lines . ........................145 iv CONTENTS 3.4 The distance formula . .........................148 3.4.1 The distance from a point to a line . .................150 4 Cevian and anticevian triangles 203 4.1 Cevian triangles . .............................203 4.1.1 The orthic triangle . .........................205 4.1.2 The intouch triangle and the Gergonne point . .........205 4.1.3 The Nagel point and the extouch triangle . .............206 4.2 The trilinear polar . .............................208 4.3 Anticevian triangles .............................209 4.3.1 The excentral triangle cev−1(I) ...................211 5 Isotomic and isogonal conjugates 213 5.1 Isotomic conjugates .............................213 5.1.1 Example: the Gergonne and Nagel points . .............214 5.1.2 Example: isotomic conjugate of the orthocenter . .........214 5.1.3 The equal-parallelians point .....................215 5.1.4 Crelle-Yff points . .........................216 5.2 Isogonal conjugates .............................218 5.2.1 The symmedian point and the centroid . .............220 5.2.2 The tangential triangle cev−1(K) ...................221 5.2.3 The Gergonne point and the insimilicenter T+ ............223 5.2.4 The Nagel point and the exsimilicenter T− ..............226 5.2.5 The Brocard points . .........................229 5.3 Isogonal conjugate of an infinite point . .................233 5.3.1 Homogeneous barycentric equation of the circumcircle . .....234 5.4 The isotomic conjugates of infinite points . .................236 6 Some basic constructions 237 6.1 Perspective triangles .............................237 6.2 Jacobi’s Theorem . .............................242 6.2.1 The Kiepert perspectors . .....................246 6.3 Gossard’s theorem . .............................247 6.4 Cevian quotients . .............................251 6.4.1 . .................................254 6.5 The cevian quotient G/P ...........................255 6.6 The cevian quotient H/P ...........................256 6.7 Pedal and reflection triangles . .....................259 6.7.1 Reflections and isogonal conjugates .................259 6.7.2 The pedal circle . .........................259 6.7.3 Pedal triangle .............................262 6.7.4 Examples . .............................262 6.7.5 Reflection triangle . .........................264 6.8 Barycentric product .............................266 6.8.1 Barycentric square . .........................266 CONTENTS v 6.8.2 Barycentric square root ........................269 7 Orthology 271 7.1 Triangle determined by orthology centers . ................271 7.1.1 Examples ...............................273 7.2 Perspective orthologic triangles . ....................275 7.2.1 . ...................................275 8 The circumcircle 303 8.1 The circumcircle ...............................303 8.1.1 Tangents to the circumcircle . ....................304 8.2 Simson lines . ...............................306 8.3 Line of reflections . ........................310 8.4 Reflections of a line in the sidelines of T ..................311 8.4.1 . ...................................314 8.4.2 Perspectivity of reflection triangles . ................316 8.5 Circumcevian triangles . ........................317 8.5.1 Circumcevian triangle ........................319 8.5.2 The circumcevian triangle of H ...................320 8.5.3 The circum-tangential triangle ....................321 8.5.4 Circumcevian triangles congruent to the reference triangle . ....322 8.6 Triangle bounded by the reflections of a tangent to the circumcircle in the sidelines ...................................323 9 Circles 325 9.1 Generic circles . ...............................325 9.2 Power of a point with respect to a circle . ................328 9.2.1 Radical axis and radical center ....................328 9.3 The tritangent circles . ........................330 9.3.1 The Feuerbach theorem ........................331 9.3.2 Radical axes of the circumcircle with the excircles . ........334 9.3.3 The radical center of the excircles . ................336 9.3.4 The Spieker radical circle . ....................337 9.4 The Conway circle . ........................339 9.4.1 Sharp’s triad of circles ........................341 9.5 The Taylor circle ...............................343 9.5.1 The Taylor circle of the excentral triangle . ............345 9.6 Some triads of circles . ........................346 9.6.1 Circles with sides as diameters ....................346 9.6.2 Circles with cevians as diameters . ................347 9.6.3 Circles with centers at vertices and altitudes as radii ........348 9.6.4 Circles with altitudes as diameters . ................349 9.6.5 Excursus: A construction problem . ................350 9.6.6 Excursus: Radical center of a triad of circles ............352 9.6.7 Concurrency of three Euler lines . ................353 vi CONTENTS 10 Tucker circles 401 10.1 The Taylor circle . .............................401 10.1.1 The pedals of pedals .........................401 10.1.2 The Taylor circle . .........................402 10.1.3 The Taylor center . .........................402 10.1.4 The Taylor circle of the excentral triangle . .............404 10.1.5 A triad of Taylor circles . .....................404 10.2 The Taylor circle . .............................405 10.2.1 . .................................406 10.3 Tucker circles . .............................408 10.3.1 The center of Tucker circle . .....................409 10.3.2 Construction of Tucker circle with given center . .........411 10.3.3 Dao’s construction of the Tucker circles . .............412 10.4 The Lemoine circles .............................414 10.4.1 The first Lemoine circle . .....................414 10.4.2 The second Lemoine circle . .....................416 10.4.3 Ehrmann’s third Lemoine circle . .................417 10.4.4 Bui’s fourth Lemoine circle .....................418 10.5 The Apollonius circle . .........................419 10.6 Tucker circles which are pedal circles . .................421 10.6.1 The Gallatly circle . .........................422 10.7 Tucker circles which are cevian circumcircles . .............423 10.7.1 Tucker circle congruent to the circumcircle .............424 10.8 Torres’ circle .................................425 10.9 Tucker circles tangent to the tritangent circles . .............426 10.9.1 The incircle . .............................426 10.9.2 The excircles .............................427 11 Some special circles 431 11.1 The Dou circle . .............................431 11.2 The Adams circles .............................433 11.3 Hagge circles . .............................434 11.3.1 . .................................434 11.4 The deLongchamps triad of circles (A(a),B(b),C(c)) ...........435 11.5 The orthial circles . .............................437 11.6 Bui’s triad of circles .............................438 11.7 Appendix: More triads of circles . .....................439 11.7.1 The triad {A(AH )} ..........................439 11.7.2 The triad of circles (AG(AH ),BG(BH ),CG(CH )) ..........442 11.8 The triad of circles (AG(AH ),BG(BH ),CG(CH )) .............443 11.9 The Lucas circles . .............................444 CONTENTS vii 12 The triangle of reflections 447 12.1 The triangle of reflections T† ........................447 12.2 Triangles with degenerate triangle of reflections . ............449 12.3 Triads of concurent circles . ........................451 12.3.1 Musselman’s theorem ........................451 12.3.2 The triad of circles AB†C†, BC†A†, CA†B†