X1159

GREENHILL POINT

X1159 = {X2099,X4860}-harmonic conjugate of X1
< 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, 3 }: OI Line (Incenter–Circumcenter Line) { 7, 952 }: Central Line L(7, 952) 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_3 = CIRCUMCENTER (0.5000, 0.1021) X_7 = GERGONNE POINT (0.5903, 0.2769) X_952 = INTERSECTION OF X(1)X(5) AND X(3)X(8) (0.1267, 0.6191) X_1159 = GREENHILL POINT (0.5934, 0.2737) A B C X1 X3 X7 X952 X1159 (a, b, c) = (5, 6, 7) | AB = 1

X1159 (GREENHILL POINT) 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 \(f(A,B,C) = \cos A + 4 \cos B + 4 \cos C\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(0.866795670955\)
▸ Incident Central Lines2 lines

{1, 3}, {7, 952}

▸ Functional Relations1 relation

Notes

See Hyacinthos #6535 and

William Gallatly, The Modern Geometry of the Triangle, 2nd edition, Hodgson, London, 1913, page 23.