X₃₁₃₀

X₁₄-CEVA CONJUGATE OF X₆

X₃₁₃₀ = isogonal conjugate of X₂₉₉₃

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

X₃₁₃₀ (X₁₄-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), {51, 61}

• Related Centers: X₁₄ (2nd ISOGONIC CENTER), X₆ (SYMMEDIAN POINT), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₆₁ (ISOGONAL CONJUGATE OF X₁₇)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₁₄ 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): \(-1.800558175496\)

▸ Incident Central Lines7 lines

{2, 3}, {6, 3457}, {14, 1606}, {16, 1495}, {51, 61}, {62, 184}, {618, 2925}

▸ Functional Relations5 relations

• X₃₁₃₀ = isogonal conjugate of X₂₉₉₃

• X₃₁₃₀ = circumcircle-inverse of X₃₂₄₆₀

• X₃₁₃₀ = X₁₄-Ceva conjugate of X₆

• X₃₁₃₀ = pole wrt circumcircle of trilinear polar of X₁₄ (line X₃₉₆X₅₂₃)

• X₃₁₃₀ = crosspoint of circumcircle intercepts of outer Napoleon circle

Notes

As a point on the Euler line, X₃₁₃₀ has Shinagawa coefficients \((3^{1/2}F + S, -3^{1/2}E - 3^{1/2}F - S)\).