X₃₅₁₆

EULER LINE INTERCEPT OF LINE X₆₄X_184

X₃₅₁₆ = homothetic center of Trinh triangle and mid-triangle of orthic and circumorthic triangles

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

X₃₅₁₆ (EULER LINE INTERCEPT OF LINE X₆₄X_184) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {64, 184}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆₄ (ISOGONAL CONJUGATE OF X₂₀), X₁₈₄ (INVERSE OF X)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 3 \cos A + \sec A\)

Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)

Search Value (6, 9, 13): \(18.108384506249\)

▸ Incident Central Lines8 lines

{2, 3}, {6, 1204}, {51, 1192}, {55, 1398}, {64, 184}, {165, 1829}, {574, 2207}, {1181, 3357}

▸ Functional Relations1 relation

• X₃₅₁₆ = homothetic center of Trinh triangle and mid-triangle of orthic and circumorthic triangles

Notes

As a point on the Euler line, X₃₅₁₆ has Shinagawa coefficients \((3F,E - 3F)\).