X₂₅₆₇

EXSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₇ is the exsimilicenter(nine-point circle, 2nd Brocard circle).

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

X₂₅₆₇ (EXSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {11, 2565}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₁₁ (FEUERBACH POINT), X₂₅₆₅ (EXSIMILICENTER)

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

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(-10.656147445034\)

▸ Incident Central Lines16 lines

{2, 3}, {11, 2565}, {12, 2566}, {485, 2560}, {486, 2561}, {1329, 2569}, {1342, 1348}, {1343, 1349}, {1670, 2039}, {1671, 2040}, {1676, 2558}, {1677, 2559}, {1698, 2572}, {1699, 2573}, {2009, 2562}, {2010, 2563}

Notes

As a point on the Euler line, X₂₅₆₇ has Shinagawa coefficients \((e - 1, e + 1)\).