X₃₂₁₀

X₅₇-CEVA CONJUGATE OF X₂

X₃₂₁₀ = anticomplement of X₃₁₂

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

X₃₂₁₀ (X₅₇-CEVA CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 312}

• Related Centers: X₂ (CENTROID), X₃₁₂ (ISOTOMIC CONJUGATE OF X₅₇)

Trilinears: \(u(-u/x + v/y + w/z) : v(u/x - v/y + w/z) : w(u/x + v/y - w/z)\), where \(x : y : z\) = X₅₇ and \(u : v : w\) = X₂

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): \(-2.812281404140\)

▸ Alternate Coordinates1 form

• Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = a^{2}(b + c) + a(b^{2} - bc + c^{2}) - bc(b + c) (E\). Danneels)

▸ Incident Central Lines13 lines

{2, 37}, {8, 38}, {43, 726}, {57, 1999}, {63, 194}, {81, 330}, {100, 1403}, {226, 1266}, {384, 2221}, {664, 1407}, {740, 982}, {3011, 3021}, {3101, 3164}

▸ Functional Relations9 relations

• X₃₂₁₀ = isotomic conjugate of isogonal conjugate of X₂₁₇₆₉

• X₃₂₁₀ = anticomplement of X₃₁₂

• X₃₂₁₀ = anticomplementary conjugate of X₂₁₂₈₆

• X₃₂₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (57, 2), (978, 3169)

• X₃₂₁₀ = crosssum of X₆₆₃ and X₁₉₇₇

• X₃₂₁₀ = polar conjugate of isogonal conjugate of X₂₀₈₀₅

• X₃₂₁₀ = perspector of Gemini triangle 36 and cross-triangle of Gemini triangles 36 and 38

• X₃₂₁₀ = complement of isogonal conjugate of X₃₆₆₁₉

• X₃₂₁₀ = complementary conjugate of complement of X₃₆₆₁₉