X582

HARMONIC OF X500

X582 = inverse-in-Brocard-circle of X500
< 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, 6 }: Brocard Axis { 283, 474 }: Central Line L(283, 474) { 517, 602 }: Central Line L(517, 602) { 212, 942 }: Central Line L(212, 942) 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_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_283 = -CROSS CONJUGATE OF X(21) (0.4475, 0.0704) X_474 = EULER LINE INTERCEPT OF LINE X(10)X(56) (0.5441, 0.2176) X_517 = ISOGONAL CONJUGATE OF X(104) (0.8180, 0.7154) X_602 = INTERSECTION OF LINES X(3)X(31) AND X(56)X(255) (0.8182, 0.6170) X_212 = -CEVA CONJUGATE OF X(41) (0.4719, 0.0786) X_942 = INCIRCLE-INVERSE OF X(36) (0.5878, 0.2633) X_582 = HARMONIC OF X(500) (0.8750, 0.7214) A B C X3 X6 X283 X474 X517 X602 X212 X942 X582 (a, b, c) = (5, 6, 7) | AB = 1

X582 (HARMONIC OF X500) 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 \(f(A,B,C) = \sin A + \cot D/2 \cos A\), cot \(D/2 = (e_{1} - e_{2} - 4abc)/[4 \operatorname{area} (a + b + c)]\), where \(e_{1} = a^{3} + b^{3} + c^{3}\), \(e_{2} = a^{2}(b + c) + b^{2}(c + a) + c^{2}(a + b)\)
Barycentrics: \((\sin A)f(A,B,C): (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-6.555367493674\)
▸ Incident Central Lines4 lines

{3, 6}, {212, 942}, {283, 474}, {517, 602}

▸ Functional Relations1 relation

  • X582 = inverse-in-Brocard-circle of X500