X₃₁₃₄

X₇₄-CEVA CONJUGATE OF X₅₂₃

X₃₁₃₄ = X₇₄-Ceva conjugate of X₅₂₃

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

X₃₁₃₄ (X₇₄-CEVA CONJUGATE OF X₅₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Circle: Nine-Point Circle (nine_pointcircle) — Passes through side midpoints M_a, M_b, M_c and altitude feet H_a, H_b, H_c; centered at X₅

• Central Lines: {5, 3154}, {2, 3} (Euler line)

• Related Centers: X₅ (NINE-POINT CENTER), X₃₁₅₄ (-CEVA CONJUGATE OF X₅₂₃), X₂ (CENTROID), 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 \(x : y : z\) = X₇₄ 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): \(3.773431632770\)

▸ Incident Central Lines3 lines

{2, 3}, {125, 526}, {2970, 2972}

▸ Functional Relations4 relations

• X₃₁₃₄ = inverse-in-nine-point circle of X₃₁₅₄

• X₃₁₃₄ = X₇₄-Ceva conjugate of X₅₂₃

• X₃₁₃₄ = crosspoint of X₂₆₄ and X₂₃₉₄

• X₃₁₃₄ = crosssum of X₁₈₄ and X₂₄₂₀

Notes

As a point on the Euler line, X₃₁₃₄ has Shinagawa coefficients \(((7E - 2F)F - 2S^{2},(E + F)(E - 2F) - 2S^{2})\).