X₁₅₉₁

POINT CHARA III

X₁₅₉₁ = complement of X₁₅₉₉

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

X₁₅₉₁ (POINT CHARA III) 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, 1599}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₁₅₉₉ (POINT CHORT I), X₃ (CIRCUMCENTER)

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

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

Search Value (6, 9, 13): \(2.166514800266\)

▸ Incident Central Lines3 lines

{2, 3}, {343, 639}, {394, 485}

▸ Functional Relations3 relations

• X₁₅₉₁ = complement of X₁₅₉₉

• X₁₅₉₁ = inverse-in-orthocentroidal-circle of X₁₅₈₃

• X₁₅₉₁ = X₁₃₀₆-Ceva conjugate of X₅₂₃

Notes

As a point on the Euler line, X₁₅₉₁ has Shinagawa coefficients \((E + S, S)\).