X₁₂₁₀

ISOGONAL CONJUGATE OF X₁₁₆₇

X₁₂₁₀ is the isogonal conjugate of X₁₁₆₇.

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

X₁₂₁₀ (ISOGONAL CONJUGATE OF X₁₁₆₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {4, 57}

• Related Centers: X₁₁₆₇ (1st SARAGOSSA POINT OF X₃₄), X₁ (INCENTER), X₂ (CENTROID), X₄ (ORTHOCENTER), X₅₇ (ISOGONAL CONJUGATE OF X₉)

Trilinears: \(1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b)\), where f(a,b,c) is as in X₁₁₆₇

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

Search Value (6, 9, 13): \(1.274488457930\)

▸ Incident Central Lines20 lines

{1, 2}, {3, 950}, {4, 57}, {5, 226}, {7, 3091}, {11, 65}, {12, 354}, {29, 58}, {36, 411}, {40, 497}, {46, 516}, {56, 515}, {79, 1156}, {142, 442}, {158, 273}, {189, 937}, {355, 999}, {381, 553}, {496, 517}, {1089, 1229}

▸ Functional Relations7 relations

• X₁₂₁₀ = isogonal conjugate of X₁₁₆₇

• X₁₂₁₀ = crosspoint of X_i and X_j for these (i,j): (2, 273), (75, 189)

• X₁₂₁₀ = crosssum of X_i and X_j for these (i,j): (6, 212), (31, 198)

• X₁₂₁₀ = complement of X₇₈

• X₁₂₁₀ = complementary conjugate of X₃₄₈₂₃

• X₁₂₁₀ = barycentric product X₁*X₁₇₈₆₂

• X₁₂₁₀ = homothetic center of 4th Euler triangle and inverse-in-incircle triangle

Notes

X₁₂₁₀ is the {X₂,X₈}-harmonic conjugate of X₉₃₆.