X₃₁₃₁

X₁₇-CEVA CONJUGATE OF X₆

X₃₁₃₁ = X₁₇-Ceva conjugate of X₆

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

X₃₁₃₁ (X₁₇-CEVA CONJUGATE OF X₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {16, 51}

• Related Centers: X₁₇ (1st NAPOLEON POINT), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₆ (2nd ISODYNAMIC POINT), X₅₁ (CENTROID OF ORTHIC TRIANGLE)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₁₇ and \(u : v : w\) = X₆

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): \(1.687159096656\)

▸ Incident Central Lines4 lines

{2, 3}, {15, 184}, {16, 51}, {17, 1607}

▸ Functional Relations4 relations

• X₃₁₃₁ = X₁₇-Ceva conjugate of X₆

• X₃₁₃₁ = circumcircle-inverse of X₃₇₉₇₄

• X₃₁₃₁ = {X₂,X₃₇₄₅₇}-harmonic conjugate of X₃₁₃₂

• X₃₁₃₁ = {X₃,X₂₅}-harmonic conjugate of X₃₁₃₂

Notes

As a point on the Euler line, X₃₁₃₁ has Shinagawa coefficients \((F - 3^{1/2}S, -E - F + 3^{1/2}S)\).