X₁₂₁₄

ISOGONAL CONJUGATE OF X₁₁₇₂

X₁₂₁₄ is the isogonal conjugate of X₁₁₇₂.

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

X₁₂₁₄ (ISOGONAL CONJUGATE OF X₁₁₇₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 3}, {2, 92}

• Related Centers: X₁₁₇₂ (1st SARAGOSSA POINT OF X₄₈), X₁ (INCENTER), X₃ (CIRCUMCENTER), X₂ (CENTROID), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

Trilinears: \(1/f(a,b,c) : 1/f(b,c,a) : 1/f(c,a,b)\), where f(a,b,c) is as in X₁₁₇₂

Barycentrics: \(\sec B + \sec C : :\)

Search Value (6, 9, 13): \(-11.761212574459\)

▸ Alternate Coordinates1 form

• Trilinears: \(\cot A (\cos B + \cos C) : \cot B (\cos C + \cos A) : \cot C (\cos A + \cos B)\) (Darij Grinberg, 4/11/03)

▸ Incident Central Lines16 lines

{1, 3}, {2, 92}, {5, 1838}, {7, 464}, {9, 223}, {10, 227}, {34, 405}, {37, 226}, {63, 77}, {72, 73}, {216, 1108}, {225, 442}, {304, 345}, {306, 307}, {333, 664}, {343, 914}

▸ Functional Relations12 relations

• X₁₂₁₄ = isogonal conjugate of X₁₁₇₂

• X₁₂₁₄ = isotomic conjugate of X₃₁₆₂₃

• X₁₂₁₄ = polar conjugate of X₁₈₉₆

• X₁₂₁₄ = trilinear pole of line X₅₂₀X₆₅₆

• X₁₂₁₄ = X₆-isoconjugate of X₂₉

• X₁₂₁₄ = X₉₂-isoconjugate of X₂₁₉₄

• X₁₂₁₄ = complement of X₉₂

• X₁₂₁₄ = X_i-Ceva conjugate of X_j for these (i,j) : (2, 226), (77, 73), (307, 72), (348, 307)

• X₁₂₁₄ = cevapoint of X_i and X_j for these (i,j): (37, 227), (71, 73)

• X₁₂₁₄ = X_i-cross conjugate of X_j for these (i,j): (71, 72), (201, 307)

• X₁₂₁₄ = crosspoint of X_i and X_j for these (i,j): (2, 63), (77, 348), (1231, 1441)

• X₁₂₁₄ = crosssum of X_i and X_j for these (i,j): (6, 19), (33, 607)