X₄₆

X₄-CEVA CONJUGATE OF X₁

X₄₆ = Bevan-circle-inverse of X₃₆

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

X₄₆ (X₄-CEVA CONJUGATE OF X₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {10, 63}

• Related Centers: X₄ (ORTHOCENTER), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₁₀ (SPIEKER CENTER), X₆₃ (ISOGONAL CONJUGATE OF X₁₉)

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

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

Tripolars: \((pending)\)

Search Value (6, 9, 13): \(-1.316070463083\)

▸ Alternate Coordinates2 forms

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

• Trilinears: \(r_{a} - R : :\), where \(r_{a}\), \(r_{b}\), \(r_{c}\) are the exradii

▸ Incident Central Lines25 lines

{1, 3}, {4, 90}, {9, 79}, {10, 63}, {19, 579}, {34, 47}, {43, 851}, {58, 998}, {78, 758}, {80, 84}, {100, 224}, {158, 412}, {169, 672}, {200, 1004}, {218, 910}, {222, 227}, {225, 254}, {226, 498}, {269, 1103}, {404, 997}, {474, 960}, {499, 946}, {595, 614}, {750, 975}, {978, 1054}

▸ Functional Relations22 relations

• X₄₆ = reflection of X_i in X_j for these (i,j): (1, 56), (1479, 1210)

• X₄₆ = isogonal conjugate of X₉₀

• X₄₆ = isotomic conjugate of X₂₀₅₇₀

• X₄₆ = circumcircle-inverse of X₃₂₇₆₀

• X₄₆ = Bevan-circle-inverse of X₃₆

• X₄₆ = X₄-Ceva conjugate of X₁

• X₄₆ = crosssum of X₃ and X₁₀₆₉

• X₄₆ = X_i-aleph conjugate of X_j for these (i,j): (4, 46), (174, 223), (188, 1079), (366, 610), (653, 1020)

• X₄₆ = X₁₀₀-beth conjugate of X₄₆

• X₄₆ = perspector of excentral and orthic triangles

• X₄₆ = orthic isogonal conjugate of X₁

• X₄₆ = excentral isogonal conjugate of X₁₄₉₀

• X₄₆ = X₂₄-of-excentral-triangle

• X₄₆ = {X₁,X₃}-harmonic conjugate of X₃₆₁₂

• X₄₆ = {X₁,X₄₀}-harmonic conjugate of X₅₁₁₉

• X₄₆ = perspector of ABC and extraversion triangle of X₉₀

• X₄₆ = trilinear product of extraversions of X₉₀

• X₄₆ = X₂₄ of reflection triangle of X₁

• X₄₆ = homothetic center of ABC and orthic triangle of reflection triangle of X₁

• X₄₆ = Cundy-Parry Phi transform of X₄₆

• X₄₆ = Cundy-Parry Psi transform of X₉₀

• X₄₆ = {X₁,X₅₇}-harmonic conjugate of X₃₃₃₈

Notes

Let J_a' be the reflection of the A-excenter in BC, and define J_b', J_c' cyclically. Let O_a be the circumcenter of AJb’J_c', and define O_b, O_c cyclically. O_aO_bO_c and ABC are perspective at X₄₆. (Randy Hutson, July 20, 2016)

Let A' be the inverse-in-Bevan-circle of the A-vertex of the hexyl triangle, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₄₆. (Randy Hutson, July 20, 2016)

Let J_aJ_bJ_c be the excentral triangle. Let A″ be the inverse-in-Bevan-circle of A, and define B″, C″ cyclically. The lines J_aA″, J_bB″, J_cC″ concur in X₄₆. (Randy Hutson, July 20, 2016)

X₄₆ is the {X₃,X₆₅}-harmonic conjugate of X₁.