X1076

POINT ALPHARD

< 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) { 3, 225 }: Central Line L(3, 225) { 4, 1038 }: Central Line L(4, 1038) { 55, 1070 }: Central Line L(55, 1070) { 56, 1072 }: Central Line L(56, 1072) { 1040, 1068 }: Central Line L(1040, 1068) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_225 = -CEVA CONJUGATE OF X(65) (0.6246, 0.4451) X_4 = ORTHOCENTER (0.6122, 0.3958) X_1038 = INTERSECTION OF LINES X(1)X(3) AND X(2)X(34) (0.5104, 0.1211) X_55 = INSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.5490, 0.1921) X_1070 = POINT ALKES (0.5823, 0.2764) X_56 = EXSIMILICENTER(CIRCUMCIRCLE, INCIRCLE) (0.6316, 0.3438) X_1072 = POINT ALNILAM (0.5532, 0.1606) X_1040 = INTERSECTION OF LINES X(1)X(3) AND X(2)X(33) (0.4854, 0.0753) X_1068 = -CROSS CONJUGATE OF X(4) (0.6478, 0.5373) X_1076 = POINT ALPHARD (0.3421, -0.3326) A B C X3 X225 X4 X1038 X55 X1070 X56 X1072 X1040 X1068 X1076 (a, b, c) = (5, 6, 7) | AB = 1

X1076 (POINT ALPHARD) 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 \(f(A,B,C) = \cos A - \cos^{2} B \cos C - \cos B \cos^{2} C\)
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): \(-6.673457311642\)
▸ Incident Central Lines5 lines

{3, 225}, {4, 1038}, {55, 1070}, {56, 1072}, {1040, 1068}