X₃₅₁₅

EULER LINE INTERCEPT OF LINE X₁₅₄X_185

X₃₅₁₅ = crosspoint of X₂₅₀ and X₁₃₀₁

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

X₃₅₁₅ (EULER LINE INTERCEPT OF LINE X₁₅₄X_185) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {154, 185}

• Related Centers: X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅), X₁₃₀₁ (COLLINGS TRANSFORM OF X₂₃₅), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₅₄ (-CEVA CONJUGATE OF X₆), X₁₈₅ (NAGEL POINT OF THE ORTHIC TRIANGLE)

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): \(2.763377383946\)

▸ Incident Central Lines12 lines

{2, 3}, {36, 1398}, {64, 1620}, {154, 185}, {159, 2929}, {165, 1902}, {187, 2207}, {232, 3053}, {1033, 1609}, {1204, 1495}, {1350, 1974}, {1452, 2646}

▸ Functional Relations4 relations

• X₃₅₁₅ = isogonal conjugate of X₁₅₀₇₇

• X₃₅₁₅ = crosspoint of X₂₅₀ and X₁₃₀₁

• X₃₅₁₅ = polar conjugate of isotomic conjugate of X₃₇₆₇₂

• X₃₅₁₅ = homothetic center of Kosnita triangle and mid-triangle of orthic and circumorthic triangles

Notes

As a point on the Euler line, X₃₅₁₅ has Shinagawa coefficients \((3F, -E - 3F)\).