X₁₅₉₃

POINT CEBALRAI

X₁₅₉₃ = orthocentroidal-circle-inverse of X₂₃₅

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

X₁₅₉₃ (POINT CEBALRAI) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 64}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₆₄ (ISOGONAL CONJUGATE OF X₂₀)

Trilinears: \(\cos A + \sec A : :\)

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

Search Value (6, 9, 13): \(-34.749038901671\)

▸ Incident Central Lines16 lines

{1, 1037}, {2, 3}, {6, 64}, {19, 1212}, {33, 56}, {34, 55}, {51, 1204}, {74, 1112}, {84, 1473}, {184, 1498}, {208, 1466}, {264, 1105}, {578, 1181}, {607, 672}, {1155, 1452}, {1208, 1471}

▸ Functional Relations15 relations

• X₁₅₉₃ = reflection of X_i in X_j for these (i,j): (4, 1595), (1181, 578)

• X₁₅₉₃ = complement of X₃₇₂₀₁

• X₁₅₉₃ = anticomplement of X₆₈₂₃

• X₁₅₉₃ = circumcircle-inverse of X₃₇₉₃₁

• X₁₅₉₃ = orthocentroidal-circle-inverse of X₂₃₅

• X₁₅₉₃ = crosspoint of X₄ and X₁₅₉₅

• X₁₅₉₃ = crosssum of X₃ and X₁₁₈₁

• X₁₅₉₃ = polar conjugate of X₃₇₈₇₄

• X₁₅₉₃ = polar-circle-inverse of complement of X₃₇₉₄₄

• X₁₅₉₃ = homothetic center of orthic triangle and reflection of tangential triangle in X₃

• X₁₅₉₃ = homothetic center of tangential triangle and reflection of orthic triangle in X₄

• X₁₅₉₃ = exsimilicenter of circumcircle and incircle of orthic triangle if ABC is acute; the insimilicenter is X₂₅

• X₁₅₉₃ = X₁₆₉₇-of-orthic-triangle if ABC is acute

• X₁₅₉₃ = {X₃,X₄}-harmonic conjugate of X₂₅

• X₁₅₉₃ = {X₁₂₁₇₁,X₁₂₁₇₂}-harmonic conjugate of X₁₂₁₆₇

Notes

As a point on the Euler line, X₁₅₉₃ has Shinagawa coefficients \((F, E - F)\).

In the plane of a triangle ABC, let

• Γ = circumcircle

• DEF = medial triangle

• E' = Γ-antipode of E

• 1. = circle that passes through E,F,E'

• A' = center of (A), and define B' and C' cyclically.

Then the triangles ABC and A'B'C' are homothetic, and their homothetic center is X₁₅₉₃. (Angel Montesdeoca, February 14, 2023)