X₃₅₄₆

EULER LINE INTERCEPT OF LINE X₄₉₈X_1038

X₃₅₄₆ = complement of X₃₀₈₉

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

X₃₅₄₆ (EULER LINE INTERCEPT OF LINE X₄₉₈X_1038) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Derived Triangle: Medial Triangle △M_aM_bM_c — vertices M_a, M_b, M_c (Formed by the midpoints of sides BC, CA, AB)

• Central Lines: {2, 3089}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₃₀₈₉ (INTERSECTION OF LINES X), X₃ (CIRCUMCENTER)

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

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

Search Value (6, 9, 13): \(4.655203932590\)

▸ Incident Central Lines8 lines

{2, 3}, {498, 1038}, {499, 1040}, {590, 1578}, {615, 1579}, {1060, 3085}, {1062, 3086}, {1092, 1899}

▸ Functional Relations1 relation

• X₃₅₄₆ = complement of X₃₀₈₉

Notes

As a point on the Euler line, X₃₅₄₆ has Shinagawa coefficients \((E - 2F, -E)\).