X₃₅₄₁

EULER LINE INTERCEPT OF LINE X₃₃X_499

X₃₅₄₁ = inverse-in-orthocentroidal-circle of X₃₅₄₂

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

X₃₅₄₁ (EULER LINE INTERCEPT OF LINE X₃₃X_499) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Anticomplementary Triangle △K_aK_bK_c — vertices K_a, K_b, K_c (Formed by lines through A, B, C parallel to opposite sides)

• Central Lines: {2, 3549}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₅₄₉ (EULER LINE INTERCEPT OF LINE X), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \sec A + 2 \sin B \sin C\)

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

Search Value (6, 9, 13): \(10.180049867191\)

▸ Incident Central Lines17 lines

{2, 3}, {33, 499}, {34, 498}, {54, 66}, {254, 264}, {317, 1078}, {393, 570}, {495, 1398}, {571, 3087}, {578, 1899}, {590, 3093}, {615, 3092}, {847, 1217}, {1092, 1352}, {1199, 2904}, {1870, 3085}, {1968, 2548}

▸ Functional Relations2 relations

• X₃₅₄₁ = inverse-in-orthocentroidal-circle of X₃₅₄₂

• X₃₅₄₁ = anticomplement of X₃₅₄₉

Notes

As a point on the Euler line, X₃₅₄₁ has Shinagawa coefficients \((2F, E)\).