X₁₂₈

X₇₄-OF-ORTHIC-TRIANGLE

X₁₂₈ = X₇₄-of-orthic triangle.

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

X₁₂₈ (X₇₄-OF-ORTHIC-TRIANGLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {137, 5}

• Related Centers: X₁₃₇ (-OF-ORTHIC-TRIANGLE), X₅ (NINE-POINT CENTER)

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = (\sec A)(\cos 2B + \cos 2C)(1 + 2 \cos 2A)(\cos 2A + 2 \cos 2B \cos 2C)\)

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): \(4.231492060175\)

X₁₂₈ lies on the nine-point circle

▸ Incident Central Lines5 lines

{5, 137}, {52, 134}, {53, 139}, {115, 233}, {125, 140}

▸ Functional Relations11 relations

• X₁₂₈ = X₇₄-of-orthic triangle.

• X₁₂₈ = reflection of X₁₃₇ in X₅

• X₁₂₈ = isogonal conjugate of X₁₅₄₀₁

• X₁₂₈ = complement of X₁₁₄₁

• X₁₂₈ = anticomplement of X₃₄₈₃₇

• X₁₂₈ = perspector of circumconic centered at X₂₃₁

• X₁₂₈ = center of circumconic that is locus of trilinear poles of lines passing through X₂₃₁

• X₁₂₈ = inverse-in-polar-circle of X₂₃₈₃

• X₁₂₈ = X₂-Ceva conjugate of X₂₃₁

• X₁₂₈ = orthojoin of X₂₃₁

• X₁₂₈ = excentral-to-ABC functional image of X₇₄