X₂₁₈

X₇-CEVA CONJUGATE OF X₅₅

X₂₁₈ = eigencenter of cevian triangle of X₇

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

X₂₁₈ (X₇-CEVA CONJUGATE OF X₅₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 6}, {3, 41}

• Related Centers: X₇ (GERGONNE POINT), X₅₅ (INSIMILICENTER), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₃ (CIRCUMCENTER), X₄₁ (-CEVA CONJUGATE OF X₃₁)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \cos^{2}(A/2) [\cos^{4}(B/2) + \cos^{4}(C/2) - \cos^{4}(A/2)]\)

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

Search Value (6, 9, 13): \(-0.293029092446\)

▸ Incident Central Lines13 lines

{1, 6}, {3, 41}, {4, 294}, {7, 277}, {32, 906}, {43, 170}, {46, 910}, {56, 101}, {65, 169}, {145, 644}, {198, 579}, {222, 241}, {279, 651}

▸ Functional Relations9 relations

• X₂₁₈ = isogonal conjugate of X₂₇₇

• X₂₁₈ = eigencenter of cevian triangle of X₇

• X₂₁₈ = eigencenter of anticevian triangle of X₅₅

• X₂₁₈ = X₇-Ceva conjugate of X₅₅

• X₂₁₈ = crosssum of X₆₅₀ and X₁₀₈₆

• X₂₁₈ = X₆₄₄-beth conjugate of X₂₁₈

• X₂₁₈ = crossdifference of every pair of points on the de Longchamps line of the intouch triangle

• X₂₁₈ = perspector of 2nd mixtilinear triangle and unary cofactor triangle of 5th mixtilinear triangle

• X₂₁₈ = perspector of excentral triangle and unary cofactor triangle of inverse-in-incircle triangle

Notes

Let A' be the center of the conic through the contact points of the B- and C- excircles with the sidelines of ABC. Define B' and C' cyclically. The triangle A'B'C' is perspective to the intouch triangle at X₂₁₈. See also X₆, X₂₅, X₂₂₂, X₉₄₀, X₁₇₄₃. (Randy Hutson, July 23, 2015)

A'B'C' is also the unary cofactor triangle of the intangents triangle. (Randy Hutson, June 7, 2019)