X₁₀₇₈

INTERSECTION OF LINES X₂X_32 AND X₃X_76

X₁₀₇₈ = X₂₄₉-Ceva conjugate of X₉₉

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

X₁₀₇₈ (INTERSECTION OF LINES X₂X_32 AND X₃X_76) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 1506}

• Related Centers: X₂ (CENTROID), X₁₅₀₆ (INSIMILICENTER OF MOSES AND NINE-POINT CIRCLES)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b^{2}c^{2} + c^{2}a^{2} + a^{2}b^{2} - a^{4})\)

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

Search Value (6, 9, 13): \(5.331930226330\)

▸ Incident Central Lines17 lines

{2, 32}, {3, 76}, {5, 316}, {24, 264}, {35, 350}, {39, 385}, {54, 69}, {140, 325}, {186, 1235}, {187, 384}, {194, 574}, {274, 404}, {298, 619}, {279, 618}, {302, 635}, {303, 636}, {7603, 7843}

▸ Functional Relations6 relations

• X₁₀₇₈ = isotomic conjugate of X₃₆₁₃

• X₁₀₇₈ = anticomplement of X₁₅₀₆

• X₁₀₇₈ = X₂₄₉-Ceva conjugate of X₉₉

• X₁₀₇₈ = complement of X₇₇₈₅

• X₁₀₇₈ = X₅₀₃₈-of-6th-Brocard-triangle

• X₁₀₇₈ = X₅₁₁₆-of-1st-anti-Brocard-triangle