X₁₉₈₅

EULER LINE INTERCEPT OF LINE X₆X_11

X₁₉₈₅ = inverse-in-orthocentroidal-circle of X₈₅₁

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

X₁₉₈₅ (EULER LINE INTERCEPT OF LINE X₆X_11) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {6, 11}

• Related Centers: X₂ (CENTROID), X₃ (CIRCUMCENTER), X₄ (ORTHOCENTER), X₆ (SYMMEDIAN POINT), X₁₁ (FEUERBACH POINT)

Trilinears: \(bc[bca^{4} + (b^{3} + c^{3})a^{3} - ua - v] : :\), where \(u = u(a,b,c) = (b + c)(b - c)^{2}(b^{2} + bc + c^{2})\) and \(v = v(a,b,c) = bc(b - c)^{2}(b + c)^{2}\)

Search Value (6, 9, 13): \(-8.515939435080\)

▸ Alternate Coordinates1 form

• Trilinears: \((\sec A + \sec B + \sec C) \csc A + (\csc A + \csc B + \csc C) \sec A : :\)

▸ Incident Central Lines6 lines

{2, 3}, {6, 11}, {42, 1837}, {184, 1746}, {1465, 1893}, {1699, 1730}

▸ Functional Relations1 relation

• X₁₉₈₅ = inverse-in-orthocentroidal-circle of X₈₅₁

Notes

As a point on the Euler line, X₁₉₈₅ has Shinagawa coefficients \((\$bcS_{B}S_{C}\$, \$bc\$S^{2})\).