X₂₅₄₂

INSIMILICENTER(BROCARD CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₂₅₄₂ (INSIMILICENTER(BROCARD CIRCLE, (X₄,2R))) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 83}, {2, 1341}

• Related Centers: X₄ (ORTHOCENTER), X₈₃ (CEVAPOINT OF CENTROID AND SYMMEDIAN POINT), X₂ (CENTROID), X₁₃₄₁ (EXSIMILICENTER)

Trilinears: \(e \cos B \cos C + \cos(A - \omega )\), where \(e = (1 - 4 \sin^{2} \omega )^{1/2}\)

Barycentrics: \(2 a^{2} (a^{4} - a^{2} b^{2} - a^{2} c^{2} - 2 b^{2} c^{2}) + (-a^{4} + b^{4} - 2 b^{2} c^{2} + c^{4}) \sqrt{a^{4} - a^{2} b^{2} + b^{4} - a^{2} c^{2} - b^{2} c^{2} + c^{4}}::\)

Search Value (6, 9, 13): \(2.261548888578\)

▸ Incident Central Lines26 lines

{2, 1341}, {4, 83}, {10, 1705}, {20, 1340}, {388, 1675}, {497, 1674}, {516, 1704}, {1349, 3091}, {1379, 14712}, {1587, 1668}, {1588, 1669}, {1678, 2551}, {1679, 2550}, {1693, 9535}, {1694, 9534}, {2011, 2545}, {2012, 2544}, {2033, 2549}, {2034, 2548}, {2039, 7797}, {2469, 2553}, {2470, 2552}, {2559, 6655}, {3557, 20088}, {6039, 7792}, {6190, 14929}

▸ Functional Relations1 relation

• X₂₅₄₂ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {4, 182, 2543}, {1341, 1348, 2}, {3618, 7790, 2543}