X₆₈

PRASOLOV POINT

Coordinates for X₆₈ can be obtained easily from the Ceva ratios given his Prasolov’s proof of concurrence.

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

X₆₈ (PRASOLOV POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {155, 5}

• Related Centers: X₁₅₅ (EIGENCENTER OF ORTHIC TRIANGLE), X₅ (NINE-POINT CENTER)

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

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

Search Value (6, 9, 13): \(1.370063903568\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines14 lines

{2, 54}, {3, 343}, {4, 52}, {5, 6}, {11, 1069}, {20, 74}, {26, 161}, {30, 64}, {65, 91}, {66, 511}, {73, 1060}, {136, 254}, {290, 315}, {568, 973}

▸ Functional Relations14 relations

• X₆₈ = reflection of X₁₅₅ in X₅

• X₆₈ = isogonal conjugate of X₂₄

• X₆₈ = isotomic conjugate of X₃₁₇

• X₆₈ = anticomplement of X₁₁₄₇

• X₆₈ = X₉₆-Ceva conjugate of X₃

• X₆₈ = cevapoint of X_i and X_j for these (i,j): (6, 161), (125, 520)

• X₆₈ = X₁₁₅-cross conjugate of X₅₂₅

• X₆₈ = pedal antipodal perspector of X₄

• X₆₈ = pedal antipodal perspector of X₁₈₆

• X₆₈ = X₆₃-isoconjugate of X₈₇₄₅

• X₆₈ = crossdifference of every pair of points on line X₉₂₄X₆₇₅₃

• X₆₈ = trilinear product of vertices of X₃-anti-altimedial triangle

• X₆₈ = orthic-to-ABC barycentric image of X₅₂

• X₆₈ = cyclocevian conjugate of X₃₄₂₈₇

Notes

Let A'B'C' be the 2nd Euler triangle. The lines AA', BB', CC' concur in X₆₈, as proved in V. V. Prasolov, Zadachi po planimetrii, Moscow, 4th edition, 2001.

Let O_a be the circle centered at the A-vertex of the orthic triangle and passing through A; define O_b and O_c cyclically. Then X₆₈ is the radical center of O_a, O_b, O_c. (Randy Hutson, November 2, 2017)

The X₃-Fuhrmann triangle is inversely similar to ABC, with similitude center X₃, and perspective to ABC at X₆₈. (Randy Hutson, November 3, 2017)