X₃₂₆₇

ISOTOMIC CONJUGATE OF X₁₁₂

X₃₂₆₇ is the isotomic conjugate of X₁₁₂.

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

X₃₂₆₇ (ISOTOMIC CONJUGATE OF X₁₁₂) 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, 2485}

• Related Centers: X₂ (CENTROID), X₂₄₈₅ (RADICAL CENTER OF {CIRCUMCIRCLE, 1ST LEMOINE CIRCLE, NINE-POINT CIRCLE})

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = (b^{2} - c^{2})(b^{2} + c^{2} - a^{2})/a^{3}\)

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

Search Value (6, 9, 13): \(13.499822794190\)

▸ Incident Central Lines4 lines

{2, 2485}, {99, 935}, {100, 2859}, {325, 523}

▸ Functional Relations8 relations

• X₃₂₆₇ = isotomic conjugate of X₁₁₂

• X₃₂₆₇ = anticomplement of X₂₄₈₅

• X₃₂₆₇ = X_i-Ceva conjugate of X_j for these (i,j): (305, 339), (670, 1502)

• X₃₂₆₇ = X_i-cross conjugate of X_j for these (i,j): (127, 2), (339,305()

• X₃₂₆₇ = crosspoint of X_i and X_j for these (i,j): (99, 1799), (670, 1502)

• X₃₂₆₇ = crosssum of X_i and X_j for these (i,j): (32, 3049), (512, 1843), (669, 1501), (2489, 3199)

• X₃₂₆₇ = crossdifference of every pair of points on the line X₃₂X₆₈₂

• X₃₂₆₇ = polar conjugate of X₃₂₇₁₃

Notes

X₃₂₆₇ is the trilinear pole of the line X₁₂₅X(339. Randy Hutson, August 15, 2013