X₉₃₃

X₄-CROSS CONJUGATE OF X₂₅₀

Let P be a point on the circumcircle, and let

(Oab) = circle through A tangent to line BC at B

(Oac) = circle through A tangent to line BC at C

B_a = point other than A where the line PB meets (Oab)

C_a = point other than A where the line PC meets (Oac)

Hab = orthocenter of APBa

Hac = orthocenter of APCa

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

X₉₃₃ (X₄-CROSS CONJUGATE OF X₂₅₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {54, 74}, {4, 137}

• Related Centers: X₄ (ORTHOCENTER), X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅), X₅₄ (KOSNITA POINT), X₇₄ (ISOGONAL CONJUGATE OF EULER INFINITY POINT), X₁₃₇ (-OF-ORTHIC-TRIANGLE)

Trilinears: \(1/(bw - cv) : :\), where \(u : v : w\) = X₅₄

Search Value (6, 9, 13): \(6.052599928400\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec A \csc(2B - 2C) : :\)

• Trilinears: \(a/[(b^{2} - c^{2})(b^{2} + c^{2} - a^{2})(b^{4} + c^{4} - a^{2}b^{2} - a^{2}c^{2} - 2b^{2}c^{2})] : :\)

X₉₃₃ lies on the circumcircle.

▸ Incident Central Lines6 lines

{4, 137}, {54, 74}, {98, 275}, {250, 476}, {270, 759}, {648, 925}

▸ Functional Relations14 relations

• X₉₃₃ = X₄-cross conjugate of X₂₅₀

• X₉₃₃ = isogonal conjugate of X₆₃₆₈

• X₉₃₃ = Λ(X₆₈₄, X₂₅₂₅); line X₆₈₄X₂₅₂₅ is the isotomic conjugate, wrt the MacBeath triangle, of the MacBeath inconic)

• X₉₃₃ = Λ(X₂₀₈₁, X₂₆₀₀) (line X₂₀₈₁X₂₆₀₀ is the trilinear polar of X₅)

• X₉₃₃ = trilinear pole of line X₆X₂₄

• X₉₃₃ = concurrence of reflections of line X₄X₅₄ in sides of ABC

• X₉₃₃ = Ψ(XI),X_j) for these (i,j): (3, 54), (4, 54), (5, 2), (6, 24), (69, 54)

• X₉₃₃ = inverse-in-polar-circle of X₁₃₇

• X₉₃₃ = X₁₅₇₇-isoconjugate of X₂₁₆

• X₉₃₃ = barycentric product X₁₁₀*X₂₇₅

• X₉₃₃ = barycentric quotient X₂₇₅/X₈₅₀

• X₉₃₃ = perspector of circumcevian triangle of X₁₈₆ and cross-triangle of ABC and circumorthic triangle

• X₉₃₃ = the point of intersection, other than A, B, and C, of the circumcircle and conic {A, B, C,PU(61)}}

• X₉₃₃ = center of the bianticevian conic of X₁ and X₄₇; i.e. the rectangular hyperbola {X₁, X₄₇, X₄₈, and vertices of their anticevian triangles}

Notes

Let A' = BC∩X₁₈₆X₅₂₃, and define B' and C' cyclically. The circumcircles of AB'C', BC'A', CA'B' concur in X₉₃₃. Note: Line X₁₈₆X₅₂₃ is the polar of X₄ wrt the circumcircle. (Randy Hutson, February 10, 2016)

Define Hbc and Hca cyclically, and define Hba and Hcb cyclically. The points Hab, Hac, Hbc, Hba, Hca, Hcb lies on a hyperbola with center = midpoint(P and H), where H = X₄, the orthocenter. This hyperbola is rectangular if and only if P = X₉₃₃. This note is related to a problem posed in 2017 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p9. For details, see El centro X₉₃₃ y una hipérbola rectangular. (Angel Montesdeoca, July 3, 2021)