X₅₂₀

ISOGONAL CONJUGATE OF X₁₀₇

As the isogonal conjugate of a point on the circumcircle, X₅₂₀ lies on the line at infinity.

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

X₅₂₀ (ISOGONAL CONJUGATE OF X₁₀₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {110, 250}, {30, 511}

• Related Centers: X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅), X₃₀ (EULER INFINITY POINT), X₅₁₁ (ISOGONAL CONJUGATE OF X₉₈)

Trilinears: \((\cos A)(\sin 2B - \sin 2C) : (\cos B)(\sin 2C - \sin 2A) : (\cos C)(\sin 2A - \sin 2B)\)

Barycentrics: \((\sin 2A)(\sin 2B - \sin 2C) : (\sin 2B)(\sin 2C - \sin 2A) : (\sin 2C)(\sin 2A - \sin 2B)\)

▸ Alternate Coordinates5 forms

• Trilinears: \((\cos A)(\tan B - \tan C) : :\)

• Barycentrics: \(\sec^{2} B - \sec^{2} C : :\)

• Barycentrics: \(\tan^{2} B - \tan^{2} C : :\)

• Barycentrics: \((\cos^{2} A)(b^{2} - c^{2}) : :\)

• Barycentrics: \(\csc 2B - \csc 2C : :\)

▸ Incident Central Lines23 lines

{6, 2435}, {30, 511}, {69, 879}, {74, 2693}, {98, 2706}, {99, 2713}, {100, 2719}, {101, 2727}, {102, 2732}, {103, 2738}, {104, 2744}, {105, 2749}, {106, 2755}, {108, 2761}, {109, 2762}, {110, 250}, {111, 2763}, {112, 2764}, {340, 850}, {647, 652}, {1364, 2632}, {2451, 2489}, {3265, 4131}

▸ Functional Relations20 relations

• X₅₂₀ = isogonal conjugate of X₁₀₇

• X₅₂₀ = complementary conjugate of X₁₂₂

• X₅₂₀ = anticomplementary conjugate of X₃₄₁₈₆

• X₅₂₀ = X_i-Ceva conjugate of X_j for these (i,j): (2, 35071), (4, 122), (68, 125), (110, 3)

• X₅₂₀ = crosspoint of X₃ and X₁₁₀

• X₅₂₀ = crosssum of X_i and X_j for these (i,j): (4, 523), (51, 647), (512, 800)

• X₅₂₀ = crossdifference of every pair of points on line X₄X₆

• X₅₂₀ = isotomic conjugate of X₆₅₂₈

• X₅₂₀ = perspector of hyperbola {A, B, C,X₂,X₃}}

• X₅₂₀ = intersection of trilinear polars of X₂ and X₃

• X₅₂₀ = X₆-isoconjugate of X₈₂₃

• X₅₂₀ = X₉₂-isoconjugate of X₁₁₂

• X₅₂₀ = trilinear pole of line X₁₆₃₆X₂₉₇₂

• X₅₂₀ = Thomson-isogonal conjugate of X₁₂₉₄

• X₅₂₀ = Lucas-isogonal conjugate of X₁₂₉₄

• X₅₂₀ = polar conjugate of X₁₅₃₅₂

• X₅₂₀ = perspector of circumconic centered at X₃₅₀₇₁

• X₅₂₀ = crosssum of X₂₄₀₀₇ and X₂₄₀₀₈ (the Kiepert hyperbola intercepts of the orthic axis)

• X₅₂₀ = pole wrt polar circle of trilinear polar of X₁₅₃₅₂ (line X₄X₅₁)

• X₅₂₀ = Cundy-Parry Phi transform of X₃₉₁₇₄