X₅₂

ORTHOCENTER OF ORTHIC TRIANGLE

X₅₂ is the orthocenter of orthic triangle.

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

X₅₂ (ORTHOCENTER OF ORTHIC TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Orthic Triangle △H_aH_bH_c — vertices H_a, H_b, H_c (Pedal triangle of Orthocenter X₄ (altitude feet))

• Concurrence Lines: H_a–X₅₂, H_b–X₅₂, H_c–X₅₂ concurring at X₅₂

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

Barycentrics: \(\tan A (\sec 2B + \sec 2C) : \tan B (\sec 2C + \sec 2A) : \tan C (\sec 2A + \sec 2B)\)

Tripolars: \((pending)\)

Search Value (6, 9, 13): \(-0.743866735147\)

▸ Alternate Coordinates3 forms

• Trilinears: \(\sec A (\sec 2B + \sec 2C) : :\)

• Trilinears: \(\cos(A - 2B) + \cos(A - 2C) : :\)

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

▸ Linear Combinations1 combination

• X₅₂ = \((r^{2} + 2rR + s^{2})\) X₁ - \(6rR\) X₂ - \((r^{2} - 4rR - 2R^{2} + s^{2})\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines11 lines

{3, 6}, {4, 68}, {5, 51}, {25, 155}, {26, 184}, {30, 185}, {49, 195}, {113, 135}, {114, 211}, {128, 134}, {129, 139}

▸ Functional Relations19 relations

• X₅₂ = reflection of X_i in X_j for these (i,j): (3, 389), (5, 143), (113, 1112), (1209, 973)

• X₅₂ = isogonal conjugate of X₉₆

• X₅₂ = isotomic conjugate of X₃₄₃₈₅

• X₅₂ = anticomplement of X₁₂₁₆

• X₅₂ = circumcircle-inverse of X₃₂₇₆₂

• X₅₂ = Brocard-circle-inverse of X₅₆₉

• X₅₂ = X_i-Ceva conjugate of X_j for these (i,j): (4, 5), (317, 467), (324, 216)

• X₅₂ = crosspoint of X₄ and X₂₄

• X₅₂ = crosssum of X₃ and X₆₈

• X₅₂ = {X₃,X₆}-harmonic conjugate of X₅₆₉

• X₅₂ = orthic isogonal conjugate of X₅

• X₅₂ = X₂₀-of-2nd Euler triangle

• X₅₂ = perspector of ABC and cross-triangle of ABC and 2nd Euler triangle

• X₅₂ = perspector of ABC and cross-triangle of ABC and Kosnita triangle

• X₅₂ = antipode of X₁₁₃ in Hatzipolakis-Lozada hyperbola

• X₅₂ = Cundy-Parry Phi transform of X₅₇₁

• X₅₂ = Cundy-Parry Psi transform of X₅₃₉₂

• X₅₂ = X₁₅₇₇-isoconjugate of X₃₂₆₉₂

• X₅₂ = excentral-to-ABC functional image of X₄

Notes

Let H_a be the foot of the A-altitude. Let B_a be the foot of the perpendicular from H_a to CA, and define C_b and A_c cyclically. Let C_a be the foot of the perpendicular from H_a to AB, and define A_b and B_c cyclically. Let A' be the orthocenter of H_aB_aC_a, and define B' and C' cyclically. The lines H_aA', H_bB', H_cC' concur in X₅₂. (Randy Hutson, December 10, 2016)

Let O_A be the circle centered at the A-vertex of the anti-Ursa-minor triangle and passing through A; define O_B and O_C cyclically. X₅₂ is the radical center of O_A, O_B, O_C. (Randy Hutson, August 30, 2020)

X₅₂ is the {X₅,X₁₄₃}-harmonic conjugate of X₅₁.