X₁₃₃

INTERSECTION OF LINES X₄X_74 AND X₅X_122

X₁₃₃ = X₁₀₆-of-orthic triangle is ABC is acute.

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

X₁₃₃ (INTERSECTION OF LINES X₄X_74 AND X₅X_122) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 107}

• Related Centers: X₄ (ORTHOCENTER), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER))

Trilinears: \((\sec A)[(\sin 2B - \sin 2C)^{2} + (\sin 2A)(\sin 2B) + (\sin 2A)(\sin 2C) - 2(\sin 2B)(\sin 2C)](2 \sin 2A - \sin 2B - \sin 2C) : :\)

Barycentrics: \((\tan A)[(\sin 2B - \sin 2C)^{2} + (\sin 2A)(\sin 2B) + (\sin 2A)(\sin 2C) - 2(\sin 2B)(\sin 2C)](2 \sin 2A - \sin 2B - \sin 2C) : :\)

Search Value (6, 9, 13): \(-2.715630711495\)

X₁₃₃ lies on the nine-point circle

▸ Incident Central Lines6 lines

{2, 1294}, {4, 74}, {5, 122}, {53, 115}, {127, 381}, {136, 235}

▸ Functional Relations13 relations

• X₁₃₃ = X₁₀₆-of-orthic triangle is ABC is acute.

• X₁₃₃ = midpoint of X₄ and X₁₀₇

• X₁₃₃ = reflection of X₁₂₂ in X₅

• X₁₃₃ = isogonal conjugate of X₁₅₄₀₄

• X₁₃₃ = complement of X₁₂₉₄

• X₁₃₃ = anticomplement of X₃₄₈₄₂

• X₁₃₃ = perspector of circumconic centered at X₁₉₉₀

• X₁₃₃ = center of circumconic that is locus of trilinear poles of lines passing through X₁₉₉₀

• X₁₃₃ = X₂-Ceva conjugate of X₁₉₉₀

• X₁₃₃ = trilinear pole wrt Euler triangle of van Aubel line

• X₁₃₃ = inverse-in-polar-circle of X₇₄

• X₁₃₃ = anticenter of cyclic quadrilateral ABCX₁₀₇

• X₁₃₃ = Λ(X₄, X₅₁), wrt orthic triangle