X505

3rd ISOSCELIZER POINT

The isoscelizer equations H(A,X)D(A,X) = H(B,X)D(B,X) = H(C,X)D(C,X) have solution X = X505. (Peter Yff, 4/9/99)
< 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) { 40, 164 }: Central Line L(40, 164) { 173, 1130 }: Central Line L(173, 1130) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_266 = ISOGONAL CONJUGATE OF X(188) (0.6074, 0.2891) X_1 = INCENTER (0.5714, 0.2333) X_40 = BEVAN POINT (0.4286, -0.0292) X_164 = INCENTER OF EXCENTRAL TRIANGLE (0.4930, 0.1115) X_173 = CONGRUENT ISOSCELIZERS POINT (0.5924, 0.2741) X_1130 = 1st STEVANOVIC POINT (0.5896, 0.2615) X_505 = 3rd ISOSCELIZER POINT (0.6161, 0.3801) A B C X266 X1 X40 X164 X173 X1130 X505 (a, b, c) = (5, 6, 7) | AB = 1

X505 (3rd ISOSCELIZER 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 \(f(A,B,C) = 1/(\sin B/2 + \sin C/2 - \sin A/2)\)
Barycentrics: \(af(A,B,C) : bf(B,C,A) : cf(C,A,B)\)
Search Value (6, 9, 13): \(-2.340839706193\)
▸ Incident Central Lines2 lines

{40, 164}, {173, 1130}

▸ Functional Relations3 relations

Notes

X505 is the perspector of ABC and the excentral triangle of the excentral triangle of ABC. (Randy Hutson, 9/23/2011)