X₂₃₁

X₂-CEVA CONJUGATE OF X₁₂₈

X₂₃₁ = radical center of cirumcircle, nine-point circle and Hutson-Parry circle

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

X₂₃₁ (X₂-CEVA CONJUGATE OF X₁₂₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 1273}

• Related Centers: X₂ (CENTROID), X₁₂₇₃ (ISOTOMIC CONJUGATE OF X₁₁₄₁)

Trilinears: u(-au + bv + cw),\(u : v : w\) = X₁₂₈

Barycentrics: \(2 a^{8} - 4 a^{6} b^{2} + 3 a^{4} b^{4} - 2 a^{2} b^{6} + b^{8} - 4 a^{6} c^{2} + 2 a^{2} b^{4} c^{2} - 4 b^{6} c^{2} + 3 a^{4} c^{4} + 2 a^{2} b^{2} c^{4} + 6 b^{4} c^{4} - 2 a^{2} c^{6} - 4 b^{2} c^{6} + c^{8} : :\)

Search Value (6, 9, 13): \(-2.059786071846\)

▸ Alternate Coordinates1 form

• Trilinears: \((\cos B) (4 \sin^{2} C - 1) (\cos A \sin B - \sin A \cos B) - (\cos C) (4 \sin^{2} B - 1) (\cos C \sin A - \sin C \cos A) : :\)

▸ Linear Combinations1 combination

• X₂₃₁ = X₃₅₇₂₇ + \(3\) X₃₈₃₉₄

X₂₃₁ lies on the cubic K752.

▸ Incident Central Lines38 lines

{2, 1273}, {3, 15551}, {4, 96}, {5, 15552}, {6, 17}, {50, 115}, {140, 570}, {216, 34833}, {230, 232}, {340, 44375}, {395, 33530}, {396, 33529}, {566, 7749}, {1609, 3517}, {1989, 2070}, {2072, 3284}, {2965, 36412}, {3001, 6036}, {3054, 33992}, {3523, 14806}, {3767, 5063}, {5158, 6639}, {5306, 37439}, {5355, 13337}, {6749, 9722}, {7735, 33872}, {7747, 9220}, {10018, 40939}, {10510, 38740}, {10619, 14533}, {11077, 16337}, {11082, 11136}, {11087, 11135}, {16534, 45938}, {32223, 46155}, {35727, 38394}, {37972, 44533}, {41770, 44134}

▸ Functional Relations18 relations

• X₂₃₁ = complement of X₁₂₇₃

• X₂₃₁ = circumcircle-inverse of X₁₅₅₅₁

• X₂₃₁ = nine-point-circle-inverse of X₁₅₅₅₂

• X₂₃₁ = complement of the isotomic conjugate of X₁₁₄₁

• X₂₃₁ = polar conjugate of the isotomic conjugate of X₅₃₉

• X₂₃₁ = X_i-complementary conjugate of X_j for these (i,j): {31, 128}, {1141, 2887}, {11077, 18589}, {46138, 21235}

• X₂₃₁ = X_i-Ceva conjugate of X_j for these (i,j): {2, 128}, {14918, 18400}

• X₂₃₁ = radical center of cirumcircle, nine-point circle and Hutson-Parry circle

• X₂₃₁ = crosspoint of X_i and X_j for these (i,j): {2, 1141}, {11082, 11087}

• X₂₃₁ = crosssum of X_i and X_j for these (i,j): {6, 1154}, {6368, 10413}, {11126, 11127}

• X₂₃₁ = crossdifference of every pair of points on line {3, 1510}

• X₂₃₁ = X₂₈₁-beth conjugate of X₂₃₀

• X₂₃₁ = perspector of circumconic centered at X₁₂₈

• X₂₃₁ = center of circumconic that is locus of trilinear poles of lines passing through X₁₂₈

• X₂₃₁ = X₆₃-isoconjugate of X₂₃₈₃

• X₂₃₁ = barycentric product X_i*X_j for these {i,j}: {4, 539}, {5, 40631}, {93, 45083}, {128, 1141}, {1263, 27423}, {3459, 10615}, {10412, 43969}

• X₂₃₁ = barycentric quotient X_i/X_j for these {i,j}: {25, 2383}, {128, 1273}, {539, 69}, {10615, 45799}, {40631, 95}, {43969, 10411}, {45083, 44180}

• X₂₃₁ = {X_i,X_j}-harmonic conjugate of X_k for these (i,j,k): {6, 2963, 233}, {17, 18, 1209}, {230, 16310, 3003}, {233, 40136, 6}, {571, 2165, 1879}, {3003, 16310, 3018}, {3767, 46262, 5063}