X₃₅₅₉

SS(sin A → cos A) OF X₃₃₃

X₃₅₅₉ = X₁₈₉₆-Ceva conjugate of X₂₉

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

X₃₅₅₉ (SS(sin A → cos A) OF X₃₃₃) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 3} (Euler line), {225, 283}

• Related Centers: X₁₈₉₆ (ZOSMA TRANSFORM OF X₁₀₄₇), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₂ (CENTROID), X₃ (CIRCUMCENTER), X₂₂₅ (-CEVA CONJUGATE OF X₆₅), X₂₈₃ (-CROSS CONJUGATE OF X₂₁)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\cos B \cos C)(\cos B + \cos C - \cos A)/(\cos B + \cos C)\)

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

Search Value (6, 9, 13): \(1.752329565067\)

▸ Incident Central Lines12 lines

{2, 3}, {58, 1785}, {90, 1896}, {158, 920}, {162, 1780}, {225, 283}, {243, 1858}, {318, 333}, {393, 1778}, {908, 1819}, {1068, 3157}, {1838, 2328}

▸ Functional Relations3 relations

• X₃₅₅₉ = X₁₈₉₆-Ceva conjugate of X₂₉

• X₃₅₅₉ = cevapoint of X_i and X_j for these (i,j): (4, 920), (46, 1068), (453, 3193)

• X₃₅₅₉ = X_i-cross conjugate of X_j for these (i,j): (46, 3193), (3193, 29)

Notes

As a point on the Euler line, X₃₅₅₉ has Shinagawa coefficients \((\$aS_{A}\$F, \$aS_{B}S_{C}\$ + (\$aS_{A}\$ - abc)F - \$a\$S^{2})\).