X₁₃₀

X₉₉-OF-ORTHIC-TRIANGLE

X(130 lies on the nine-point circle

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

X₁₃₀ (X₉₉-OF-ORTHIC-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {5, 129}, {51, 138}

• Related Centers: X₅ (NINE-POINT CENTER), X₁₂₉ (-OF-ORTHIC-TRIANGLE), X₅₁ (CENTROID OF ORTHIC TRIANGLE), X₁₃₈ (-OF-ORTHIC-TRIANGLE)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sin A)(\sin 2B + \sin 2C)[(\sin 2B - \sin 2C)^{2}][\sin^{2}(2A) + \sin 2B \sin 2C]\)

Barycentrics: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = (\sin A) f(A,B,C)\)

Search Value (6, 9, 13): \(-2.850328504280\)

▸ Incident Central Lines2 lines

{5, 129}, {51, 138}

▸ Functional Relations10 relations

• X₁₃₀ = X₉₉-of-orthic triangle

• X₁₃₀ = center of the rectangular hyperbola that passes through A, B, C, and X₅₁

• X₁₃₀ = reflection of X₁₂₉ in X₅

• X₁₃₀ = complement of X₁₃₀₃

• X₁₃₀ = anticomplement of X₃₄₈₃₉

• X₁₃₀ = crosssum of circumcircle intercepts of line X₃X₉₅

• X₁₃₀ = excentral-to-ABC functional image of X₉₉

• X₁₃₀ = trilinear pole wrt orthic triangle of line X₅₁X₅₃

• X₁₃₀ = orthopole of line X₃X₉₅

• X₁₃₀ = Kirikami-six-circles image of X₅₁