X569

HARMONIC OF X52

X569 = inverse-in-Brocard-circle of X52
< 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 { 2, 54 }: Central Line L(2, 54) { 26, 51 }: Central Line L(26, 51) { 5, 156 }: Central Line L(5, 156) { 140, 343 }: Central Line L(140, 343) 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_2 = CENTROID (0.5374, 0.2000) X_54 = KOSNITA POINT (0.5880, 0.2176) X_26 = CIRCUMCENTER OF THE TANGENTIAL TRIANGLE (0.1382, -0.8447) X_51 = CENTROID OF ORTHIC TRIANGLE (0.6093, 0.3186) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_156 = -OF-TANGENTIAL-TRIANGLE (0.6520, -0.1707) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_343 = ISOTOMIC CONJUGATE OF X(275) (0.4345, 0.0893) X_569 = HARMONIC OF X(52) (0.5651, 0.2096) A B C X3 X6 X2 X54 X26 X51 X5 X156 X140 X343 X569 (a, b, c) = (5, 6, 7) | AB = 1

X569 (HARMONIC OF X52) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sin A + \cot D/2 \cos A : :\)
Search Value (6, 9, 13): \(2.078407331492\)
▸ Alternate Coordinates1 form

  • Trilinears: \(\cot D/2 = (2e_{3} + e_{2} - e_{1})/[64 (\operatorname{area})^{3}]\), where \(e_{1} = a^{6} + b^{6} + c^{6}\), \(e_{2} = a^{4}(b^{2} + c^{2}) + b^{4}(c^{2} + a^{2}) + c^{4}(a^{2} + b^{2})\), \(e_{3} = a^{2}b^{2}c^{2}\)

▸ Incident Central Lines5 lines

{2, 54}, {3, 6}, {5, 156}, {26, 51}, {140, 343}

▸ Functional Relations1 relation

  • X569 = inverse-in-Brocard-circle of X52