X2672

2nd GOLDEN ARBELOS POINT

In the construction of X2671, if A' is the center of the s-Archimedean circle of the inward arbelos on segment BC, and B' and C' are defined cyclically, then A'A, B'B, C'C concur in X2672.
< 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) { 6, 2671 }: Central Line L(6, 2671) { 2, 2673 }: Central Line L(2, 2673) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2674 = ISOGONAL CONJUGATE OF X(2672) (0.6690, 0.3812) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_2671 = 1st GOLDEN ARBELOS POINT (0.5720, 0.2475) X_2 = CENTROID (0.5374, 0.2000) X_2673 = ISOGONAL CONJUGATE OF X(2671) (0.5710, 0.2193) X_2672 = 2nd GOLDEN ARBELOS POINT (0.3508, 0.0923) A B C X2674 X6 X2671 X2 X2673 X2672 (a, b, c) = (5, 6, 7) | AB = 1

X2672 (2nd GOLDEN ARBELOS POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/(2 \cos A - τ^{2}\sin A) : :\), where τ = (1 + \(\sqrt{5}\))/2 = golden ratio
Barycentrics: \((S+(\sqrt{5}-3) S_{B}) \cdot (S+(\sqrt{5}-3) S_{C}) : :\) (C. Lozada, 7/13/19)
Search Value (6, 9, 13): \(0.008052583970\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\sec[A + arccot(3 - 5^{1/2})]\) (M. Iliev, 5/13/07)

X2672 lies on the Kiepert hyperbola.

▸ Incident Central Lines2 lines

{2, 2673}, {6, 2671}

▸ Functional Relations1 relation