X1128

2nd DE VILLIERS POINT

X1128 = isogonal conjugate of X1130
< 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) { 164, 173 }: Central Line L(164, 173) { 188, 519 }: Central Line L(188, 519) { 258, 505 }: Central Line L(258, 505) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1130 = 1st STEVANOVIC POINT (0.5896, 0.2615) X_164 = INCENTER OF EXCENTRAL TRIANGLE (0.4930, 0.1115) X_173 = CONGRUENT ISOSCELIZERS POINT (0.5924, 0.2741) X_188 = 2nd MID-ARC POINT OF ANTICOMPLEMENTARY TRIANGLE (0.5245, 0.1802) X_519 = ISOGONAL CONJUGATE OF X(106) (0.9566, 0.6107) X_258 = CONGRUENT INCIRCLES ISOSCELIZER POINT (0.5464, 0.1945) X_505 = 3rd ISOSCELIZER POINT (0.6161, 0.3801) X_1128 = 2nd DE VILLIERS POINT (0.5507, 0.2059) A B C X1130 X164 X173 X188 X519 X258 X505 X1128 (a, b, c) = (5, 6, 7) | AB = 1

X1128 (2nd DE VILLIERS 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 \(f(A,B,C) = (\sin((\pi - A)/4))/\sin((\pi + 3A)/4)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(2.307985300852\)
▸ Alternate Coordinates1 form

  • Trilinears: \(gA,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = [1 - 2 \sin(A/2)]/(1 - 2 \cos A)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines3 lines

{164, 173}, {188, 519}, {258, 505}

▸ Functional Relations1 relation

Notes

Let A″,B″,C″ be the excenters of ABC, and let A', B', C' be the respective incenters of triangles A″BC, B″CA, C″AB, respectively. The triangle A'B'C' is perspective to ABC, and the perspector is X1128. (Darij Grinberg, 8/22/02). See references at X1127.