X₂₂₀

X₉-CEVA CONJUGATE OF X₅₅

X₂₂₀ = cevapoint of X₁ and X₁₇₀

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

X₂₂₀ (X₉-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}, {41, 55}

• Related Centers: X₉ (MITTENPUNKT), X₅₅ (INSIMILICENTER), X₁ (INCENTER), X₆ (SYMMEDIAN POINT), X₄₁ (-CEVA CONJUGATE OF X₃₁)

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

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

Search Value (6, 9, 13): \(4.040506245816\)

▸ Alternate Coordinates1 form

• Trilinears: \((1 + \cos A)^{2}/\sin A : :\) (M. Iliev, 4/12/07)

▸ Incident Central Lines20 lines

{1, 6}, {3, 101}, {8, 294}, {33, 210}, {40, 910}, {41, 55}, {48, 963}, {63, 241}, {64, 71}, {78, 949}, {144, 279}, {154, 205}, {169, 517}, {200, 728}, {201, 221}, {268, 577}, {277, 1086}, {281, 594}, {329, 948}, {346, 1043}

▸ Functional Relations11 relations

• X₂₂₀ = isogonal conjugate of X₂₇₉

• X₂₂₀ = anticomplement of X₂₁₂₅₈

• X₂₂₀ = X_i-Ceva conjugate of X_j for these (i,j): (9, 55), (200, 480)

• X₂₂₀ = cevapoint of X₁ and X₁₇₀

• X₂₂₀ = crosspoint of X₉ and X₂₀₀

• X₂₂₀ = crosssum of X₅₇ and X₂₆₉

• X₂₂₀ = crossdifference of every pair of points on line X₅₁₃X₆₇₆

• X₂₂₀ = X_i-beth conjugate of X_j for these (i,j): (101, 221), (220, 41), (644, 220), (728, 728)

• X₂₂₀ = {X₁,X₉}-harmonic conjugate of X₁₂₁₂

• X₂₂₀ = perspector of ABC and unary cofactor triangle of inverse-in-incircle triangle

• X₂₂₀ = barycentric square of X₉

Notes

The trilinear polar of X₂₂₀ passes through X₆₅₇ (Randy Hutson, July 20, 2016)

Let A' be the barycentric product of the circumcircle intercepts of the A-excircle. Define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₂₀. (Randy Hutson, July 11, 2019)