X1531

ORTHOJOIN OF X32

X1531 = orthopole of de Longchamps line
< 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) { 4, 69 }: Central Line L(4, 69) { 30, 113 }: Central Line L(30, 113) { 382, 1092 }: Central Line L(382, 1092) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_69 = SYMMEDIAN POINT OF THE ANTICOMPLEMENTARY TRIANGLE (0.4122, 0.0654) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_113 = JERABEK ANTIPODE (0.4319, 0.4718) X_382 = REFLECTION OF CIRCUMCENTER IN ORTHOCENTER (0.7245, 0.6894) X_1092 = TRILINEAR CUBE OF X(3) (0.3503, 0.0118) X_1531 = ORTHOJOIN OF X(32) (0.0445, -0.5419) A B C X4 X69 X30 X113 X382 X1092 X1531 (a, b, c) = (5, 6, 7) | AB = 1

X1531 (ORTHOJOIN OF X32) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \([2a^{4} - a^{2}(b^{2} + c^{2}) - b^{4} - c^{4} + 2b^{2}c^{2}][a^{5} - a(b^{2} - c^{2})^{2} - 4bc(b^{3} \cos B + c^{3} \cos C)] : :\)
Search Value (6, 9, 13): \(-3.696054664938\)
▸ Incident Central Lines3 lines

{4, 69}, {30, 113}, {382, 1092}

▸ Functional Relations2 relations

Notes

Let MaMbMc be the Ehrmann mid-triangle. Let A' be the crosspoint of Mb and Mc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1531. (Randy Hutson, July 31 2018)