X₃₁₄₃

X₁₁₁-CEVA CONJUGATE OF X₅₂₃

X₃₁₄₃ = X₁₁₁-Ceva conjugate of X₅₂₃

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

X₃₁₄₃ (X₁₁₁-CEVA CONJUGATE OF X₅₂₃) 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, 804}

• Related Centers: X₁₁₁ (PARRY POINT), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₈₀₄ (ODD (- 2, 2) INFINITY POINT)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₁₁₁ and \(u : v : w\) = X₅₂₃

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): \(4.351254431093\)

▸ Incident Central Lines4 lines

{2, 3}, {115, 804}, {125, 2780}, {339, 2971}

▸ Functional Relations2 relations

• X₃₁₄₃ = X₁₁₁-Ceva conjugate of X₅₂₃

• X₃₁₄₃ = crossdifference of every pair of points on the line X₆₄₇X₁₆₃₄

Notes

As a point on the Euler line, X₃₁₄₃ has Shinagawa coefficients \((2(E + F)^{2}F - (E - 2F)S^{2},2(E + F)^{3} - (7E - 2F)S^{2})\).