X₁₂₇₀

ISOTOMIC CONJUGATE OF X₁₁₃₁

X₁₂₇₀ is the isotomic conjugate of X₁₁₃₁.

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

X₁₂₇₀ (ISOTOMIC CONJUGATE OF X₁₁₃₁) 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, 3068}

• Related Centers: X₂ (CENTROID), X₃₀₆₈ (INTERSECTION OF X)

Trilinears: \(1/(a^{2}f(a,b,c)) : 1/(b^{2}f(b,c,a)) : 1/(c^{2}f(c,a,b))\), where f(a,b,c) is as in X₁₁₃₁)

Search Value (6, 9, 13): \(4.816485839496\)

▸ Alternate Coordinates1 form

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A, B, C) = (\csc A)(1 + 2 \cot A)\)

▸ Incident Central Lines6 lines

{2, 6}, {4, 1160}, {8, 175}, {20, 488}, {76, 1132}, {320, 1267}

▸ Functional Relations5 relations

• X₁₂₇₀ = isotomic conjugate of X₁₁₃₁

• X₁₂₇₀ = anticomplement of X₃₀₆₈

• X₁₂₇₀ = X₁₁₅₁-cross conjugate of X₃₅₃₅

• X₁₂₇₀ = homothetic center of anticomplementary triangle and cross-triangle of ABC and outer Grebe triangle

• X₁₂₇₀ = homothetic center of outer Grebe triangle and cross-triangle of ABC and outer Grebe triangle

Notes

X₁₂₇₀ is the {X₂,X₆₉}-harmonic conjugate of X₁₂₇₁.