X₃₅₃₆

EULER LINE INTERCEPT OF LINE X₁₉X_587

X₃₅₃₆ = inverse-in-orthocentroidal-circle of X₃₅₃₅

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

X₃₅₃₆ (EULER LINE INTERCEPT OF LINE X₁₉X_587) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {19, 587}

• Related Centers: X₁₁₅₂ (ISOGONAL CONJUGATE OF X₁₁₃₂), X₁₂₇₁ (ISOTOMIC CONJUGATE OF X₁₁₃₂), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₁₉ (CLAWSON POINT), X₅₈₇ (POINT ARCTURUS)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sec A - 2 \csc A\)

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

Search Value (6, 9, 13): \(1.857562904837\)

▸ Incident Central Lines8 lines

{2, 3}, {19, 587}, {275, 3316}, {393, 615}, {590, 3087}, {1249, 3069}, {1870, 3084}, {2052, 3317}

▸ Functional Relations3 relations

• X₃₅₃₆ = inverse-in-orthocentroidal-circle of X₃₅₃₅

• X₃₅₃₆ = X₁₁₅₂-cross conjugate of X₁₂₇₁

• X₃₅₃₆ = polar conjugate of X₁₁₃₂

Notes

As a point on the Euler line, X₃₅₃₆ has Shinagawa coefficients \((2F, -S)\).