X₁₃₁

INTERSECTION OF LINES X₃X_125 AND X₄X_135

X₁₃₁ = X₁₀₂-of-orthic triangle if ABC is acute.

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

X₁₃₁ (INTERSECTION OF LINES X₃X_125 AND X₄X_135) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {136, 5}

• Related Centers: X₁₃₆ (INTERSECTION OF LINE X), X₅ (NINE-POINT CENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec A)[2T - S(\sec 2B + \sec 2C)](T - S \sec 2A)\), \(S = \sin 2A + \sin 2B + \sin 2C\), \(T = \tan 2A + \tan 2B + \tan 2C\)

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): \(-3.113862158337\)

X₁₃₁ lies on the nine-point circle

▸ Incident Central Lines4 lines

{3, 125}, {4, 135}, {5, 136}, {115, 216}

▸ Functional Relations7 relations

• X₁₃₁ = X₁₀₂-of-orthic triangle if ABC is acute.

• X₁₃₁ = reflection of X₁₃₆ in X₅

• X₁₃₁ = complement of X₁₃₀₀

• X₁₃₁ = anticomplement of X₃₄₈₄₀

• X₁₃₁ = complementary conjugate of X₁₃₇₅₄

• X₁₃₁ = inverse-in-polar-circle of X₁₂₉₉

• X₁₃₁ = Λ(X₄, X₅₂), wrt orthic triangle