X₁₃₄₈

INSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)

X₁₃₄₈ is the insimilicenter(nine-point circle, Brocard circle).

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

X₁₃₄₈ (INSIMILICENTER(NINE-POINT CIRCLE, BROCARD CIRCLE)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 182}, {2, 1341}

• Related Centers: X₅ (NINE-POINT CENTER), X₁₈₂ (MIDPOINT OF BROCARD DIAMETER), X₂ (CENTROID), X₁₃₄₁ (EXSIMILICENTER)

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

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

Search Value (6, 9, 13): \(2.385787721856\)

▸ Alternate Coordinates1 form

• Trilinears: \(e \cos A + 2 e \cos B \cos C + \cos(A - \omega ) : :\)

▸ Linear Combinations4 combinations

• X₁₃₄₈ = \(|OK|\) X₅ + \(R\) X₁₈₂

• X₁₃₄₈ = \((e \sec \omega )\) X₅ + X₁₈₂

• X₁₃₄₈ = X₃ + \(2(e \sec \omega )\) X₅ + X₆

• X₁₃₄₈ = \(3\) X₂ + X₃₅₉₁₃

▸ Incident Central Lines38 lines

{2, 1341}, {3, 2040}, {4, 1340}, {5, 182}, {6, 2039}, {10, 1694}, {11, 1674}, {12, 1675}, {115, 2033}, {316, 1379}, {485, 1668}, {486, 1669}, {574, 19660}, {1329, 1678}, {1342, 2567}, {1343, 2566}, {1346, 2470}, {1347, 2469}, {1380, 38227}, {1506, 2034}, {1664, 5403}, {1665, 5404}, {1679, 2886}, {1693, 2051}, {1698, 1705}, {1699, 1704}, {2009, 2012}, {2010, 2011}, {2543, 3091}, {2558, 37446}, {2559, 5025}, {3414, 7694}, {3557, 7785}, {6039, 9993}, {6177, 7746}, {7818, 39022}, {7828, 14631}, {13414, 13870}

▸ Functional Relations3 relations

• X₁₃₄₈ = internal center of similitude of nine-point circle and Brocard circle (Peter J. C. Moses, 4/2003)

• X₁₃₄₈ = crosssum of X₁₃₄₁ and X₃₅₅₇

• X₁₃₄₈ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {2, 2542, 1341}, {5, 182, 1349}, {3589, 7844, 1349}