X₄₆₉

EULER LINE INTERCEPT OF LINE X₉₂X_264

X₄₆₉ = inverse-in-orthocentroidal-circle of X₂₇

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

X₄₆₉ (EULER LINE INTERCEPT OF LINE X₉₂X_264) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {92, 264}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT), X₂₆₄ (ISOTOMIC CONJUGATE OF CIRCUMCENTER)

Trilinears: \(u \csc 2A : v \csc 2B : w \csc 2C\), where \(u : v : w\) = X₃₈₆; e.g., \(u(a,b,c) = a(b^{2} + c^{2} + bc + ca + ab)\)

Barycentrics: \(u \sec A : v \sec B : w \sec C\)

Search Value (6, 9, 13): \(-17.594579768507\)

▸ Incident Central Lines3 lines

{2, 3}, {92, 264}, {226, 273}

▸ Functional Relations1 relation

• X₄₆₉ = inverse-in-orthocentroidal-circle of X₂₇

Notes

As a point on the Euler line, X₄₆₉ has Shinagawa coefficients \((F,E + F + \$bc\$)\).

X₄₆₉ is the {X₂,X₄}-harmonic conjugate of X₂₇.