X3645

POINT ALKAID

Denote the incenter and excenters by I, IA, IB, IC. Let JA be the incenter of triangle BCI, and define JB and JC cyclically. The lines IAJA, IBJB, ICJC concur in X3645.
< 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) { 1, 168 }: Central Line L(1, 168) { 40, 483 }: Central Line L(40, 483) { 258, 1127 }: Central Line L(258, 1127) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_168 = MITTENPUNKT OF EXCENTRAL TRIANGLE (0.4565, 0.0098) X_40 = BEVAN POINT (0.4286, -0.0292) X_483 = RADICAL CENTER OF AJIMA-MALFATTI CIRCLES (0.5762, 0.2431) X_258 = CONGRUENT INCIRCLES ISOSCELIZER POINT (0.5464, 0.1945) X_1127 = 1st DE VILLIERS POINT (0.5779, 0.2467) X_3645 = POINT ALKAID (0.5812, 0.2523) A B C X1 X168 X40 X483 X258 X1127 X3645 (a, b, c) = (5, 6, 7) | AB = 1

X3645 (POINT ALKAID) 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 \(f(A,B,C) = 1 + \cos(B/2) + \cos(C/2) - \cos(A/2)\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) ; (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.292001690467\)

X3645 lies on the Euler line of the BCI triangle. (Randy Hutson, January 29, 2018)

▸ Incident Central Lines3 lines

{1, 168}, {40, 483}, {258, 1127}