X₁₁₁₂

CROSSPOINT OF X₄ AND X₂₅₀

X₁₁₁₂ is the crosspoint of X₄ and X₂₅₀.

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

X₁₁₁₂ (CROSSPOINT OF X₄ AND X₂₅₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {974, 389}

• Related Centers: X₉₇₄ (2nd EHRMANN POINT), X₃₈₉ (CENTER OF THE TAYLOR CIRCLE)

Trilinears: \(a[a^{4}(b^{2} + c^{2}) - 2a^{2}(b^{4} + c^{4}) + b^{6} + c^{6} ]/(b^{2} + c^{2} - a^{2}) : :\)

Search Value (6, 9, 13): \(-1.564197476780\)

▸ Alternate Coordinates1 form

• Trilinears: \(\sec A (1 + \cos 3A \cos(B - C)) : :\)

▸ Incident Central Lines8 lines

{4, 94}, {6, 1177}, {25, 110}, {51, 125}, {52, 113}, {389, 974}, {428, 542}, {468, 511}

▸ Functional Relations8 relations

• X₁₁₁₂ = reflection of X₉₇₄ in X₃₈₉

• X₁₁₁₂ = crosspoint of X₄ and X₂₅₀

• X₁₁₁₂ = crosssum of X₃ and X₁₂₅

• X₁₁₁₂ = inverse-in-polar-circle of X₃₄₄₈

• X₁₁₁₂ = polar conjugate of isotomic conjugate of X₃₄₉₉₀

• X₁₁₁₂ = inverse-in-orthosymmedial-circle of X₄₂₇

• X₁₁₁₂ = excentral-to-ABC functional image of X₁₁

• X₁₁₁₂ = intersection of tangents to Walsmith rectangular hyperbola at X₁₁₀ and X₁₂₅

Notes

X₁₁₁₂ is the center of the hyperbola that passes through the vertices of the cevian triangles of X₄ and X₆₄₈, and also through the centers X_i for I = 4, 113, 155, 193. (Paul Yiu, Oct. 16, 2001, as contributing editor for Clark Kimberling, “Conics associated with a cevian nest,” :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 141-150; see Example 2.)

X₁₁₁₂ is X₁₁-of-the-orthic-triangle if ABC is acute. (Peter Moses, July 7, 2009)