X1056

POINT ALCYONE

X1056 = mixtilinear-excentral-to-mixtilinear-incentral similarity image of X4
< 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, 4 }: Incenter–Orthocenter Line { 30, 390 }: Central Line L(30, 390) { 55, 376 }: Central Line L(55, 376) { 8, 443 }: Central Line L(8, 443) { 2, 495 }: Central Line L(2, 495) { 145, 377 }: Central Line L(145, 377) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1057 = ISOGONAL CONJUGATE OF X(1056) (0.5671, 0.2170) X_1 = INCENTER (0.5714, 0.2333) X_4 = ORTHOCENTER (0.6122, 0.3958) X_30 = EULER INFINITY POINT (0.7469, 0.7481) X_390 = REFLECTION OF GERGONNE POINT IN INCENTER (0.5526, 0.1897) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_376 = CENTROID OF THE ANTIPEDAL TRIANGLE OF X(2) (0.4626, 0.0042) X_8 = NAGEL POINT (0.4694, 0.1333) X_443 = COMPLEMENT OF X(452) (0.5427, 0.2138) X_2 = CENTROID (0.5374, 0.2000) X_495 = JOHNSON MIDPOINT (0.5666, 0.2382) X_145 = ANTICOMPLEMENT OF NAGEL POINT (0.6735, 0.3333) X_377 = EULER LINE INTERCEPT OF LINE X(7)X(8) (0.5473, 0.2258) X_1056 = POINT ALCYONE (0.5756, 0.2499) A B C X1057 X1 X4 X30 X390 X55 X376 X8 X443 X2 X495 X145 X377 X1056 (a, b, c) = (5, 6, 7) | AB = 1

X1056 (POINT ALCYONE) 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 \(f(A,B,C) = 2 + \cos B \cos C\)
Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A) f(A,B,C)\)
Search Value (6, 9, 13): \(1.293169538008\)
▸ Linear Combinations1 combination
▸ Incident Central Lines13 lines

{1, 4}, {2, 495}, {7, 517}, {8, 443}, {29, 1059}, {30, 390}, {55, 376}, {56, 631}, {145, 377}, {329, 392}, {355, 938}, {529, 1001}, {1037, 1065}

▸ Functional Relations2 relations