X₈₆₈

CROSSPOINT OF X₉₈ AND X₅₂₃

X₈₆₈ is the crosspoint of X₉₈ and X₅₂₃.

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

X₈₆₈ (CROSSPOINT OF X₉₈ AND X₅₂₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {115, 125}

• Related Centers: X₉₈ (TARRY POINT), X₅₂₃ (ISOGONAL CONJUGATE OF X₁₁₀), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₂₅ (CENTER OF JERABEK HYPERBOLA)

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

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

Search Value (6, 9, 13): \(3.867614707249\)

▸ Alternate Coordinates1 form

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

▸ Incident Central Lines3 lines

{2, 3}, {115, 125}, {127, 136}

▸ Functional Relations8 relations

• X₈₆₈ = crosspoint of X₉₈ and X₅₂₃

• X₈₆₈ = crosssum of X₁₁₀ and X₅₁₁

• X₈₆₈ = crossdifference of every pair of points on line X₁₁₀X₆₄₇

• X₈₆₈ = X₁₁₅-Hirst inverse of X₁₂₅

• X₈₆₈ = intersection of trilinear polar and polar wrt polar circle of X₆₄₈

• X₈₆₈ = inverse-in-Hutson-Parry-circle of X₁₂₅

• X₈₆₈ = {X₁₃₆₃₆,X₁₃₇₂₂}-harmonic conjugate of X₁₂₅

• X₈₆₈ = complementary conjugate of complement of X₃₅₃₆₄

Notes

As a point on the Euler line, X₈₆₈ has Shinagawa coefficients \((3(E + F)F - S^{2},(E + F)^{2} - 3S^{2})\).