X₅₂₁

ISOGONAL CONJUGATE OF X₁₀₈

As the isogonal conjugate of a point on the circumcircle, X₅₂₁ lies on the line at infinity.

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

X₅₂₁ (ISOGONAL CONJUGATE OF X₁₀₈) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {59, 100}, {30, 511}

• Related Centers: X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER)), X₅₉ (ISOGONAL CONJUGATE OF X₁₁), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

Trilinears: \((\sec B - \sec C)(\csc A) : (\sec C - \sec A)(\csc B) : (\sec A - \sec B)(\csc C)\)

Barycentrics: \(\sec B - \sec C : \sec C - \sec A : \sec A - \sec B\)

▸ Alternate Coordinates3 forms

• Trilinears: \((b - c)(b + c - a)(b^{2} + c^{2} - a^{2}) : :\)

• Trilinears: directed distance from A to line X₁X₄ : :

• Barycentrics: \((\cos A)(\cos B - \cos C) : :\)

▸ Incident Central Lines18 lines

{6, 2509}, {30, 511}, {59, 100}, {74, 2694}, {98, 2707}, {99, 2714}, {101, 2728}, {102, 2733}, {103, 2739}, {104, 2745}, {109, 2765}, {110, 2766}, {111, 2767}, {650, 1021}, {651, 677}, {656, 810}, {1364, 2968}, {4025, 4131}

▸ Functional Relations15 relations

• X₅₂₁ = isogonal conjugate of X₁₀₈

• X₅₂₁ = isotomic conjugate of X₁₈₀₂₆

• X₅₂₁ = complementary conjugate of X₁₂₃

• X₅₂₁ = anticomplementary conjugate of X₃₄₁₈₈

• X₅₂₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 35072), (4, 123), (100, 3)

• X₅₂₁ = perspector of hyperbola {A, B, C,X₂,X₂₁}}

• X₅₂₁ = crosspoint of X₈ and X₁₀₀

• X₅₂₁ = crosssum of X_i and X_j for these (i,j): (33, 650), (56, 513), (429, 523), (663, 1400)

• X₅₂₁ = crossdifference of every pair of points on line X₆X₁₉

• X₅₂₁ = polar conjugate of isogonal conjugate of X₃₆₀₅₄

• X₅₂₁ = X₆-isoconjugate of X₆₅₃

• X₅₂₁ = X₉₂-isoconjugate of X₁₄₁₅

• X₅₂₁ = Thomson-isogonal conjugate of X₁₂₉₅

• X₅₂₁ = Lucas-isogonal conjugate of X₁₂₉₅

• X₅₂₁ = Cundy-Parry Phi transform of X₃₉₁₇₅