X₃₅₇₅

EULER LINE INTERCEPT OF THE LINE X₆₄X_66

X₃₅₇₅ = reflection of X₁₈₈₅ in X₄

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

X₃₅₇₅ (EULER LINE INTERCEPT OF THE LINE X₆₄X_66) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1885, 4}, {2, 3} (Euler line)

• Related Centers: X₁₈₈₅ (ZOSMA TRANSFORM OF X₇₇₄), X₄ (ORTHOCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc[2a^{6} + b^{6} + c^{6} - 3a^{4}b^{2} - 3a^{4}c^{2} - b^{4}c^{2} - b^{2}c^{4}]/(b^{2} + c^{2} - a^{2})\)

Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)

Search Value (6, 9, 13): \(3.044799668126\)

▸ Incident Central Lines15 lines

{1, 1892}, {2, 3}, {53, 571}, {64, 66}, {128, 135}, {185, 1503}, {225, 1852}, {317, 1975}, {515, 1829}, {516, 1902}, {1179, 1300}, {1192, 1853}, {1452, 1837}, {1828, 2829}, {1862, 1872}

▸ Functional Relations11 relations

• X₃₅₇₅ = reflection of X₁₈₈₅ in X₄

• X₃₅₇₅ = circumcircle-inverse of X₃₇₉₅₄

• X₃₅₇₅ = X₄-Ceva conjugate of X₃₅₇₄

• X₃₅₇₅ = X₆₅-of-orthic-triangle if ABC is acute

• X₃₅₇₅ = Kosnita-to-orthic similarity image of X₅

• X₃₅₇₅ = {X₄,X₂₄}-harmonic conjugate of X₅

• X₃₅₇₅ = intersection of tangents to Hatzipolakis-Lozada hyperbola at X₄ and X₅₂

• X₃₅₇₅ = crosspoint, wrt orthic triangle, of X₄ and X₅₂

• X₃₅₇₅ = Ehrmann-vertex-to-orthic similarity image of X₅

• X₃₅₇₅ = excentral-to-ABC functional image of X₆₅

• X₃₅₇₅ = X₄-of-anti-Ara-triangle

Notes

As a point on the Euler line, X₃₅₇₅ has Shinagawa coefficients \((F, -E - 3F)\).

Let KH denote the hyperbola discussed at X₁₁₁₂. As noted in the paper cited at X₁₁₁₂, KH is the X₄-Ceva conjugate of the Euler line. Inversely, the Euler line is the X₄-Ceva conjugate of KH. Since X₃₅₇₄ lies on KH, its X₄-Ceva conjugate, which is X₃₅₇₅, lies on the Euler line. (Peter Moses, July 7, 2009)