X₂₅₃

X₄-CROSS CONJUGATE OF X₂

X₂₅₃ is the perspector of ABC and the pedal triangle of X₆₄.

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

X₂₅₃ (X₄-CROSS CONJUGATE OF X₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {20, 64}, {7, 280}

• Related Centers: X₄ (ORTHOCENTER), X₂ (CENTROID), X₂₀ (DE LONGCHAMPS POINT), X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₇ (GERGONNE POINT), X₂₈₀ (-CROSS CONJUGATE OF X₈)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\csc A)/(\tan B + \tan C - \tan A)\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 1/(\tan B + \tan C - \tan A)\)

Search Value (6, 9, 13): \(6.547843480141\)

▸ Alternate Coordinates2 forms

• Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\csc^{2} A)/(\cos A - \cos B \cos C)\)

• Barycentrics: \(1/(3 a^{4} - 2 a^{2} (b^{2} + c^{2}) - (b^{2} - c^{2})^{2}) : :\)

X₂₅₃ lies on the Lucas cubic.

▸ Incident Central Lines8 lines

{2, 1073}, {7, 280}, {8, 307}, {20, 64}, {193, 287}, {306, 329}, {318, 342}, {322, 341}

▸ Functional Relations10 relations

• X₂₅₃ = isogonal conjugate of X₁₅₄

• X₂₅₃ = isotomic conjugate of X₂₀

• X₂₅₃ = cyclocevian conjugate of X₆₉

• X₂₅₃ = cevapoint of X_i and X_j for these (i,j): (4, 459), (122, 525)

• X₂₅₃ = X_i-cross conjugate of X_j for these (i,j): (4, 2), (122, 525)

• X₂₅₃ = anticomplement of X₁₂₄₉

• X₂₅₃ = polar conjugate of X₁₂₄₉

• X₂₅₃ = perspector of ABC and the reflection in X₁₀₇₃ of the pedal triangle of X₁₀₇₃

• X₂₅₃ = perspector of de Longchamps circle

• X₂₅₃ = pole, wrt de Longchamps circle, of trilinear polar of X₆₉ (line X₄₄₁X₅₂₅)

Notes

X₂₅₃ is the unique point whose complement is also its polar conjugate (X₁₂₄₉). (Randy Hutson, July 11, 2019)