X1509

ISOGONAL CONJUGATE OF X1500

X1509 is the isogonal conjugate of X1500.
< 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) { 1, 99 }: Central Line L(1, 99) { 58, 86 }: Central Line L(58, 86) { 81, 239 }: Central Line L(81, 239) { 76, 940 }: Central Line L(76, 940) { 552, 553 }: Central Line L(552, 553) { 2, 1171 }: Central Line L(2, 1171) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1500 = INSIMILICENTER OF MOSES CIRCLE AND INCIRCLE (0.5747, 0.2319) X_1 = INCENTER (0.5714, 0.2333) X_99 = STEINER POINT (0.1109, 0.4322) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_86 = CEVAPOINT OF INCENTER AND CENTROID (0.5534, 0.2171) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_239 = -LINE CONJUGATE OF X(42) (4.2109, 3.7992) X_76 = 3rd BROCARD POINT (0.4567, 0.1388) X_940 = POINT ALDEBARAN (0.5757, 0.2411) X_552 = POINT MAIA I (0.6112, 0.3127) X_553 = POINT MAIA II (0.5963, 0.2857) X_2 = CENTROID (0.5374, 0.2000) X_1171 = 1st SARAGOSSA POINT OF X(42) (0.6052, 0.2763) X_1509 = ISOGONAL CONJUGATE OF X(1500) (0.5682, 0.2347) A B C X1500 X1 X99 X58 X86 X81 X239 X76 X940 X552 X553 X2 X1171 X1509 (a, b, c) = (5, 6, 7) | AB = 1

X1509 (ISOGONAL CONJUGATE OF X1500) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/[a(b + c)^{2}] : 1/[b(c + a)^{2}] : 1/[c(a + b)^{2}]\)
Barycentrics: \(1/(b + c)^{2} : 1/(c + a)^{2} : 1/(a + b)^{2}\)
Search Value (6, 9, 13): \(1.756100305907\)
▸ Incident Central Lines7 lines

{1, 99}, {2, 1171}, {58, 86}, {76, 940}, {81, 239}, {552, 553}, {593, 763}

▸ Functional Relations6 relations

Notes

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″ be the trilinear pole of the tangent to the circumcircle at A', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur at X1509. (Randy Hutson, June 27, 2018)