X₂₅₉₆

CROSSDIFFERENCE OF PU(68)

X₂₅₉₆ = X₂₆₀₃-Ceva conjugate of X₂₅₉₅

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

X₂₅₉₆ (CROSSDIFFERENCE OF PU(68)) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 5}, {54, 1087}

• Related Centers: X₂₆₀₃ (TRILINEAR POLE OF PU(69)), X₂₅₉₅ (CROSSSUM OF PU(68)), X₁ (INCENTER), X₅ (NINE-POINT CENTER), X₅₄ (KOSNITA POINT), X₁₀₈₇ (TRILINEAR SQUARE OF X₅)

Trilinears: \(\cos(A - B) \cos(A - C) - \cos^{2}(B - C) : :\)

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

Search Value (6, 9, 13): \(0.421829626340\)

▸ Incident Central Lines4 lines

{1, 5}, {54, 1087}, {215, 2607}, {654, 1021}

▸ Functional Relations5 relations

• X₂₅₉₆ = isogonal conjugate of X₂₅₉₇

• X₂₅₉₆ = X₂₆₀₃-Ceva conjugate of X₂₅₉₅

• X₂₅₉₆ = crosssum of X₂₆₀₁ and X₂₆₀₂

• X₂₅₉₆ = {X₅₄,X₁₀₈₇}-harmonic conjugate of X₂₅₉₅

• X₂₅₉₆ = perspector of conic {{A, B, C,PU(69),X₆₅₅,X₃₆₁₅}}

Notes

Let L = X₁X₅, the trilinear polar of X₆₅₅. Let M = X₆₅₄X₁₀₁₂, the trilinear polar of the cevapoint of X₁ and X₅. Let V = X₃₆₁₅ = X₁-Ceva conjugate of X₅ and W = X₃₄₆₀ = X₅-Ceva conjugate of X₁. The lines L, M, an VW concur in X₂₅₉₆. (Randy Hutson, December 26, 2015)