X3649

KS(INTOUCH TRIANGLE)

Two proofs of Dao’s theorem about cyclic hexagons were published in 2014: Nikolaos Dergiades’s Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon and Telv Cohl’s A Purely Synthetic Proof of Dao’s Theorem on Six Circumcenters Associated with a Cyclic Hexagon. In the degenerate case that the hexagon is the intouch triangle A'B'C', the triangle of the circumcenters of AB'C', A'BC', A'B'C is perspective to A'B'C' with perspector X3649. (Dao Thanh Oai, Francisco Javier Garcia Capitan, ADGEOM #1718, September 16, 2014)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) incircle: Incircle of △ABC { 10, 12 }: Central Line L(10, 12) { 7, 21 }: Central Line L(7, 21) { 1, 30 }: Central Line L(1, 30) { 11, 113 }: Central Line L(11, 113) { 57, 191 }: Central Line L(57, 191) { 145, 388 }: Central Line L(145, 388) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_12 = {X(1),X(5)}-HARMONIC CONJUGATE OF X(11) (0.5641, 0.2407) X_7 = GERGONNE POINT (0.5903, 0.2769) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_113 = JERABEK ANTIPODE (0.4319, 0.4718) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_191 = -CEVA CONJUGATE OF X(1) (0.4859, 0.1209) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_388 = ∩X(7)X(8) (0.5790, 0.2635) X_3649 = KS(INTOUCH TRIANGLE) (0.5845, 0.2676) A B C X1 X10 X12 X7 X21 X30 X11 X113 X57 X191 X145 X388 X3649 (a, b, c) = (5, 6, 7) | AB = 1

X3649 (KS(INTOUCH TRIANGLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b + c)(2a + b + c)(a - b + c)(a + b - c)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(0.868611407718\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b + c)(2a + b + c)(a - b + c)(a + b - c)\)

▸ Incident Central Lines26 lines

{1, 30}, {7, 21}, {10, 12}, {11, 113}, {55, 3487}, {57, 191}, {140, 3336}, {145, 388}, {354, 946}, {396, 2306}, {429, 1835}, {553, 1125}, {940, 1406}, {962, 3303}, {1056, 2098}, {1086, 1193}, {1100, 1839}, {1317, 1365}, {1358, 2795}, {1360, 1367}, {1411, 2647}, {1749, 3337}, {1834, 2650}, {1852, 1870}, {1901, 2294}, {3318, 3324}

▸ Functional Relations2 relations

Notes

Let A' be the midpoint of A and X1, and define B' and C' cyclically. Triangle A'B'C' is perspective to the intouch triangle at X3649. (Randy Hutson, June 7, 2019)