X₄₆₃

EULER LINE INTERCEPT OF LINE X₅₁X_398

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

X₄₆₃ (EULER LINE INTERCEPT OF LINE X₅₁X_398) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {51, 398}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₃₉₈ (CROSSPOINT OF X₄ AND X₁₈)

Trilinears: \(u \tan A : v \tan B : w \tan C\), where \(u : v : w\) = X₃₉₆; e.g., \(u(A,B,C) = \cos(B - C) + 2 \cos(A - \pi /3)\)

Barycentrics: \(u \sin A \tan A : v \sin B \tan B : w \sin C \tan C\)

Search Value (6, 9, 13): \(-1.200468744346\)

▸ Incident Central Lines3 lines

{2, 3}, {51, 398}, {184, 397}

Notes

As a point on the Euler line, X₄₆₃ has Shinagawa coefficients \((3^{1/2}F, -3^{1/2}(E + F) - 2S)\).