X₃₁₅₀

X₂₉₀-CEVA CONJUGATE OF X₅₂₅

X₃₁₅₀ = crossdifference of X_i and X_j for every pair of points on the line X₆₄₇X₁₆₂₄

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

X₃₁₅₀ (X₂₉₀-CEVA CONJUGATE OF X₅₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {115, 122}

• Related Centers: X₂₉₀ (ISOGONAL CONJUGATE OF X₂₃₇), X₅₂₅ (ISOGONAL CONJUGATE OF X₁₁₂), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₂₂ (-OF-MEDIAL-TRIANGLE)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₂₉₀ and \(u : v : w\) = X₅₂₅

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): \(3.711100452250\)

▸ Incident Central Lines5 lines

{2, 3}, {115, 122}, {125, 127}, {339, 2972}, {879, 2435}

▸ Functional Relations3 relations

• X₃₁₅₀ = X_i-Ceva conjugate of X_j for these (i,j): (290, 525), (1297, 523)

• X₃₁₅₀ = crossdifference of X_i and X_j for every pair of points on the line X₆₄₇X₁₆₂₄

• X₃₁₅₀ = nine-point-circle-inverse of X₃₇₉₈₇

Notes

As a point on the Euler line, X₃₁₅₀ has Shinagawa coefficients \((2(E + F)(2E - F)F - (E + 2F)S^{2}, -2(E + F)^{2}F + (E - 2F)S^{2})\).