X1130

1st STEVANOVIC POINT

X1130 = isogonal conjugate of X1128
< 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, 164 }: Central Line L(1, 164) { 173, 505 }: Central Line L(173, 505) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1128 = 2nd DE VILLIERS POINT (0.5507, 0.2059) X_1 = INCENTER (0.5714, 0.2333) X_164 = INCENTER OF EXCENTRAL TRIANGLE (0.4930, 0.1115) X_173 = CONGRUENT ISOSCELIZERS POINT (0.5924, 0.2741) X_505 = 3rd ISOSCELIZER POINT (0.6161, 0.3801) X_1130 = 1st STEVANOVIC POINT (0.5896, 0.2615) A B C X1128 X1 X164 X173 X505 X1130 (a, b, c) = (5, 6, 7) | AB = 1

X1130 (1st STEVANOVIC POINT) 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 \(f(A,B,C) = \sin[(\pi + 3A)/4]/\sin[(\pi - A)/4]\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.141141543674\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 1 + 2 \sin(A/2)\)

▸ Incident Central Lines2 lines

{1, 164}, {173, 505}

▸ Functional Relations1 relation

Notes

Let U be the A-excenter of triangle ABC; let A' be the incenter of triangle UBC, and define B', C' cyclically. Let A″ = BC'∩CB', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X1130. (Milorad R. Stevanovic, Hyacinthos #7185, 5/21/03. See also X1488 and X1489.)