X₃₃₁₀

BARYCENTRIC PRODUCT X₅₁₃*X₅₁₇

X₃₃₁₀ = crossdifference of every pair of points on the line X₃X₈

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

X₃₃₁₀ (BARYCENTRIC PRODUCT X₅₁₃*X₅₁₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {230, 231}, {6, 654}

• Related Centers: X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈), X₆ (SYMMEDIAN POINT), X₆₅₄ (CROSSDIFFERENCE OF X₁ AND X₅)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a(b - c)( -1 + \cos B + \cos C)\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(-1.339463545513\)

▸ Incident Central Lines8 lines

{6, 654}, {37, 1639}, {42, 926}, {230, 231}, {244, 665}, {649, 854}, {909, 2423}, {1024, 1945}

▸ Functional Relations13 relations

• X₃₃₁₀ = isogonal conjugate of X₁₃₁₃₆

• X₃₃₁₀ = X_i-Ceva conjugate of X_j for these (i,j): (2, 3259), (1411, 3271), (2397, 517), (2401, 513), (2427, 2183)

• X₃₃₁₀ = crosspoint of X_i and X_j for these (i,j): (2, 901), (106, 1461), (513, 2401), (517, 2397), (650, 2432), (2183, 2427)

• X₃₃₁₀ = crosssum of X_i and X_j for these (i,j): (6, 900), (100, 2427), (104, 2423), (519, 3239), (651, 2406)

• X₃₃₁₀ = crossdifference of every pair of points on the line X₃X₈

• X₃₃₁₀ = polar conjugate of isogonal conjugate of X₂₃₂₂₀

• X₃₃₁₀ = polar conjugate of isotomic conjugate of X₈₆₇₇

• X₃₃₁₀ = complement of complement of polar conjugate of isogonal conjugate of X₂₃₁₈₄

• X₃₃₁₀ = X₆₃-isoconjugate of X₁₃₀₉

• X₃₃₁₀ = perspector of hyperbola {A, B, C,X₄,X₅₆} (circumconic centered at X₃₂₅₉ and the isogonal conjugate of line X₃X₈)

• X₃₃₁₀ = center of circumconic that is locus of trilinear poles of lines passing through X₃₂₅₉

• X₃₃₁₀ = intersection of trilinear polars of X₄ and X₅₆

• X₃₃₁₀ = barycentric product of circumcircle intercepts of Sherman line