X₉₀

X₃-CROSS CONJUGATE OF X₁

X₉₀ = isogonal conjugate of X₄₆

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

X₉₀ (X₃-CROSS CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {9, 35}, {4, 46}

• Related Centers: X₃ (CIRCUMCENTER), X₁ (INCENTER), X₉ (MITTENPUNKT), X₃₅ ({X), X₄ (ORTHOCENTER), X₄₆ (-CEVA CONJUGATE OF X₁)

Trilinears: \(1/(\cos B + \cos C - \cos A) : 1/(\cos C + \cos A - \cos B) : 1/(\cos A + \cos B - \cos C)\)

Barycentrics: \(a/(\cos B + \cos C - \cos A) : b/(\cos C + \cos A - \cos B) : c/(\cos A + \cos B - \cos C)\)

Search Value (6, 9, 13): \(2.722260516138\)

▸ Alternate Coordinates1 form

• Trilinears: \(1/(a^{3} + a^{2} (b + c) - a (b^{2} + c^{2}) - (b - c)^{2} (b + c)) : :\)

▸ Linear Combinations1 combination

• X₉₀ = \((r + R)^{2}\) X₁ - \(6rR\) X₂ - \(2r(r - R)\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines8 lines

{1, 155}, {4, 46}, {9, 35}, {21, 224}, {33, 47}, {36, 84}, {40, 80}, {57, 79}

▸ Functional Relations8 relations

• X₉₀ = isogonal conjugate of X₄₆

• X₉₀ = isotomic conjugate of X₂₀₉₃₀

• X₉₀ = X₃-cross conjugate of X₁

• X₉₀ = perspector of ABC and extraversion triangle of X₄₆

• X₉₀ = trilinear product of the extraversions of X₄₆, which is also the cross-triangle of the orthic and excentral triangles

• X₉₀ = trilinear product of PU(125)

• X₉₀ = Cundy-Parry Phi transform of X₉₀

• X₉₀ = Cundy-Parry Psi transform of X₄₆