X₁₄₄₁

X₃₁₃-BETH CONJUGATE OF X₃₁₃

X₁₄₄₁ = isotomic conjugate of X₂₁

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

X₁₄₄₁ (X₃₁₃-BETH CONJUGATE OF X₃₁₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 8}, {2, 92}

• Related Centers: X₃₁₃ (ISOTOMIC CONJUGATE OF X₅₈), X₇ (GERGONNE POINT), X₈ (NAGEL POINT), X₂ (CENTROID), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (1 - \cos A)u(a,b,c)\), where \(u : v : w\) = X₃₁₃

Barycentrics: \((\csc A) (\cos B + \cos C) : :\)

Search Value (6, 9, 13): \(1.777123010153\)

▸ Alternate Coordinates1 form

• Barycentrics: \(b c (b + c)/(a - b - c) : :\)

▸ Incident Central Lines19 lines

{2, 92}, {7, 8}, {10, 307}, {12, 313}, {19, 379}, {21, 286}, {34, 964}, {57, 1150}, {86, 664}, {95, 404}, {226, 306}, {253, 318}, {269, 996}, {274, 961}, {287, 651}, {305, 561}, {443, 1119}, {1074, 1111}, {1402, 1447}

▸ Functional Relations9 relations

• X₁₄₄₁ = isogonal conjugate of X₂₁₉₄

• X₁₄₄₁ = isotomic conjugate of X₂₁

• X₁₄₄₁ = complement of polar conjugate of isogonal conjugate of X₂₃₁₇₁

• X₁₄₄₁ = X_i-Ceva conjugate of X_j for these (i,j): (75, 307), (85, 226), (349, 321), (664, 693)

• X₁₄₄₁ = X_i-cross conjugate of X_j for these (i,j): (10, 321), (12, 226), (226, 1446), (442, 2), (121, 76), (1214, 1231)

• X₁₄₄₁ = cevapoint of X_i and X_j for these (i,j): (10, 226), (65, 1214)

• X₁₄₄₁ = crosspoint of X₇₅ and X₂₆₄

• X₁₄₄₁ = crosssum of X₃₁ and X₁₈₄

• X₁₄₄₁ = polar conjugate of X₁₁₇₂