X₁₅₉₉

POINT CHORT I

X₁₅₉₉ = anticomplement of X₁₅₉₁

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

X₁₅₉₉ (POINT CHORT I) 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, 1591}, {2, 3} (Euler line)

• Related Centers: X₂ (CENTROID), X₁₅₉₁ (POINT CHARA III), X₃ (CIRCUMCENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = \csc A + 2 \cos A\)

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

Search Value (6, 9, 13): \(3.555076776934\)

▸ Incident Central Lines3 lines

{2, 3}, {6, 588}, {394, 1151}

▸ Functional Relations2 relations

• X₁₅₉₉ = anticomplement of X₁₅₉₁

• X₁₅₉₉ = X₁₅₈₅-Ceva conjugate of X₁₉₉₃

Notes

As a point on the Euler line, X₁₅₉₉ has Shinagawa coefficients \((E + 2S, -2S)\).