X2676

(EULER LINE)∩X2671X_2672

< 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 { 6, 2671 }: Central Line L(6, 2671) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2671 = 1st GOLDEN ARBELOS POINT (0.5720, 0.2475) X_2676 = (EULER LINE)∩X(2671)X(2672) (0.5496, 0.2318) A B C X2 X3 X4 X6 X2671 X2676 (a, b, c) = (5, 6, 7) | AB = 1

X2676 ((EULER LINE)∩X2671X_2672) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(16R^{2}\cos B \cos C + (3τ + 2)bc : :\), where τ = (1 + \(\sqrt{5}\))/2 = golden ratio
Barycentrics: \(6 (S^{2}+S_{B} S_{C}) \sqrt{5}+13 S^{2}+2 S_{B} S_{C} : :\) (C. Lozada, 7/13/19)
Search Value (6, 9, 13): \(1.276728981283\)
▸ Alternate Coordinates1 form

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

▸ Incident Central Lines2 lines

{2, 3}, {6, 2671}

Notes

As a point on the Euler line, X2676 has Shinagawa coefficients \((h - 1, -h - 1)\).

Contributed by Peter J. C. Moses, August, 31, 2004.