X₂₆₂₉

CROSSSUM PU(74)

X₂₆₂₉ = X₆₅₆-Ceva conjugate of X₁

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

X₂₆₂₉ (CROSSSUM PU(74)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 162}, {191, 1047}

• Related Centers: X₆₅₆ (CROSSDIFFERENCE OF X₁ AND X₁₉), X₁ (INCENTER), X₁₆₂ (CEVAPOINT OF X₁₀₈ AND X₁₀₉), X₁₉₁ (-CEVA CONJUGATE OF X₁), X₁₀₄₇ (-CEVA CONJUGATE OF X₁)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\tan A - \tan B)(\tan A - \tan C) - (\tan B - \tan C)^{2}\),

Barycentrics: \(a (a^{8} - a^{6} b^{2} - 2 a^{4} b^{4} + 3 a^{2} b^{6} - b^{8} - a^{6} c^{2} + 5 a^{4} b^{2} c^{2} - 3 a^{2} b^{4} c^{2} - b^{6} c^{2} - 2 a^{4} c^{4} - 3 a^{2} b^{2} c^{4} + 4 b^{4} c^{4} + 3 a^{2} c^{6} - b^{2} c^{6} - c^{8}) : :\)

Search Value (6, 9, 13): \(41.883101398240\)

X₂₆₂₉ lies on the conic TCX₁₀₅₄.

▸ Incident Central Lines16 lines

{1, 162}, {9, 2634}, {63, 1956}, {191, 1047}, {293, 1707}, {610, 2247}, {656, 9390}, {896, 1955}, {1046, 1771}, {1490, 5974}, {1768, 2636}, {1954, 6508}, {2631, 16562}, {2939, 9324}, {9355, 21381}, {9359, 16575}

▸ Functional Relations5 relations

• X₂₆₂₉ = isogonal conjugate of X₉₃₉₀

• X₂₆₂₉ = X₆₅₆-Ceva conjugate of X₁

• X₂₆₂₉ = X₂₆₃₃-cross conjugate of X₁

• X₂₆₂₉ = X_i-aleph conjugate of X_j for these (i,j): (1, 656), (2, 1577)

• X₂₆₂₉ = crosspoint of PU(75)