X₃₄₄₅

X₅₆-VERTEX CONJUGATE OF X₅₆

X₃₄₄₅ = isogonal conjugate of X₁₄₅

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

X₃₄₄₅ (X₅₆-VERTEX CONJUGATE OF X₅₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 106}, {1, 474}

• Related Centers: X₁₄₅ (ANTICOMPLEMENT OF NAGEL POINT), X₃ (CIRCUMCENTER), X₁₀₆ (Λ(INCENTER, CENTROID)), X₁ (INCENTER), X₄₇₄ (EULER LINE INTERCEPT OF LINE X)

Trilinears: \(a/(3a - b - c) : b/(3b - c - a) : c/(3c - a - b)\)

Barycentrics: \(a^{2}/(3a - b - c) : b^{2}/(3b - c - a) : c^{2}/(3c - a - b)\)

Search Value (6, 9, 13): \(-8.693015191493\)

▸ Incident Central Lines23 lines

{1, 474}, {2, 1222}, {3, 106}, {6, 1201}, {8, 1120}, {34, 1319}, {56, 1149}, {58, 999}, {87, 1001}, {220, 1015}, {244, 2098}, {269, 1279}, {937, 1104}, {958, 979}, {995, 1126}, {996, 1125}, {1220, 2899}, {1357, 2308}, {1388, 1411}, {1413, 1457}, {2176, 2279}, {2191, 2646}, {2256, 2983}

▸ Functional Relations11 relations

• X₃₄₄₅ = perspector of ABC and the inner-mixtilinear tangents triangle (see X₁₁₀₅₁.

• X₃₄₄₅ = isogonal conjugate of X₁₄₅

• X₃₄₄₅ = isotomic conjugate of complement of polar conjugate of isogonal conjugate of X₂₃₁₈₂

• X₃₄₄₅ = anticomplement of X₂₈₈₅

• X₃₄₄₅ = cevapoint of X₆₆₃ and X₁₀₁₅

• X₃₄₄₅ = X_i-cross conjugate of X_j for these (i,j): (55, 6), (1357, 513), (2347, 57)

• X₃₄₄₅ = crosspoint of X₁ and X₂₁₃₇

• X₃₄₄₅ = crosssum of X_i and X_j for these (i,j): (1, 2136), (1743, 3158)

• X₃₄₄₅ = trilinear pole of line X₆₄₉X₆₃₆₃

• X₃₄₄₅ = perspector of ABC and unary cofactor of triangle T(-2,1)

• X₃₄₄₅ = crossdifference of every pair of points on line X₂₉₇₆X₃₆₆₇ (the radical axis of incircle and AC-incircle)

Notes

Let Tan_aTan_bTan_c be the tangential triangle of ABC, and let L be the line through X₁ parallel to BC. Let A″ = L∩Tan_bTan_c, and define B″ and C″ cyclically. Let A* = Tan_bB″∩Tan_cC″, and define B* and C* cyclically. The lines AA*, BB*, CC* concur in X₃₄₄₅; also, the lines Tan_aA*, Tan_bB*, Tan_cC* concur in X₃₀₅₂. (Angel Montesdeoca, April 29, 2016)