X₆₈₆

CROSSDIFFERENCE OF X₄ AND X₁₁₀

X₆₈₆ = perspector of hyperbola {A, B, C,X₃,X₅₂₃}}

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

X₆₈₆ (CROSSDIFFERENCE OF X₄ AND X₁₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {115, 125}, {184, 351}

• Related Centers: X₂ (CENTROID), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₂₅ (CENTER OF JERABEK HYPERBOLA), X₁₈₄ (INVERSE OF X), X₃₅₁ (CENTER OF THE PARRY CIRCLE)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sec B \csc(A - B) + \sec C \csc(A - C)\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = \sin A f(A,B,C)\)

Search Value (6, 9, 13): \(-2.171958801121\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines3 lines

{115, 125}, {184, 351}, {520, 647}

▸ Functional Relations5 relations

• X₆₈₆ = isogonal conjugate of X₆₈₇

• X₆₈₆ = X₂-Ceva conjugate of X₃₉₀₀₅

• X₆₈₆ = perspector of hyperbola {A, B, C,X₃,X₅₂₃}}

• X₆₈₆ = crossdifference of every pair of points on line X₄X₁₁₀

• X₆₈₆ = areal center of cevian triangles of X₇₄ and X₁₁₀