X₃₁₅₃

X₃₂₈-CEVA CONJUGATE OF X₂

X₃₁₅₃ = anticomplement of X₁₈₆

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

X₃₁₅₃ (X₃₂₈-CEVA CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 186}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄), X₃ (CIRCUMCENTER)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₃₂₈ and \(u : v : w\) = X₂

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\), where \(f(a,b,c) = u(-u/x + v/y + w/z)\)

Search Value (6, 9, 13): \(0.880666863788\)

▸ Incident Central Lines6 lines

{2, 3}, {110, 1568}, {146, 1531}, {265, 1154}, {827, 2697}, {1287, 1297}

▸ Functional Relations6 relations

• X₃₁₅₃ = refection of X_i in X_j for these (i,j): (20, 2071), (23, 403), (110, 1568), (186, 2072), (2070, 5), (2071, 858)

• X₃₁₅₃ = anticomplement of X₁₈₆

• X₃₁₅₃ = inverse-in-de-Longchamps-circle of X₃

• X₃₁₅₃ = inverse-in-Johnson-circle of X₅

• X₃₁₅₃ = homothetic center of dual of orthic triangle and Ehrmann vertex-triangle

• X₃₁₅₃ = X₄₈₄-of-Ehrmann-vertex-triangle

Notes

As a point on the Euler line, X₃₁₅₃ has Shinagawa coefficients \((E, -2E - 8F)\).