X2675

X2671X_2673∩X2672X_2674

This point of intersection and the fact that it lies on the Euler line were contributed by Peter J. C. Moses, August, 31, 2004.
< 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) { 2, 3 }: Euler Line { 2671, 2673 }: Central Line L(2671, 2673) { 2672, 2674 }: Central Line L(2672, 2674) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_4 = ORTHOCENTER (0.6122, 0.3958) X_2671 = 1st GOLDEN ARBELOS POINT (0.5720, 0.2475) X_2673 = ISOGONAL CONJUGATE OF X(2671) (0.5710, 0.2193) X_2672 = 2nd GOLDEN ARBELOS POINT (0.3508, 0.0923) X_2674 = ISOGONAL CONJUGATE OF X(2672) (0.6690, 0.3812) X_2675 = ∩X(2672)X(2674) (0.5736, 0.2945) A B C X2 X3 X4 X2671 X2673 X2672 X2674 X2675 (a, b, c) = (5, 6, 7) | AB = 1

X2675 (X2671X_2673∩X2672X_2674) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(16R^{2}(\cos B \cos C - \cos A) + (3τ + 2)bc : :\), where τ = (1 + \(\sqrt{5}\))/2 = golden ratio
Barycentrics: \(12 (S^{2}+S_{B} S_{C}) \sqrt{5}+7 S^{2}+68 S_{B} S_{C} : :\) (C. Lozada, 7/13/19)
Search Value (6, 9, 13): \(-1.382627244388\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cos B \cos C - h \cos A\), where h = [5 - 4(\(5^{1/2}\)]/15 (M. Iliev, 5/13/07)

▸ Incident Central Lines3 lines

{2, 3}, {2671, 2673}, {2672, 2674}

Notes

As a point on the Euler line, X2675 has Shinagawa coefficients \((h, -h - 1)\).