X₁₃₂

X₂X_107∩X₄X_32

X₁₃₂ = midpoint of X₄ and X₁₁₂

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

X₁₃₂ (X₂X_107∩X₄X_32) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 112}

• Related Centers: X₄ (ORTHOCENTER), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT))

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

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): \(-2.938541066207\)

X₁₃₂ lies on the nine-point circle.

▸ Incident Central Lines8 lines

{2, 107}, {4, 32}, {5, 127}, {25, 136}, {51, 125}, {114, 684}, {137, 428}, {147, 648}

▸ Functional Relations19 relations

• X₁₃₂ = midpoint of X₄ and X₁₁₂

• X₁₃₂ = reflection of X₁₂₇ in X₅

• X₁₃₂ = isogonal conjugate of X₁₅₄₀₇

• X₁₃₂ = anticomplement of X₃₄₈₄₁

• X₁₃₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 232), (4, 1503)

• X₁₃₂ = X₄-line conjugate of X₂₄₈

• X₁₃₂ = crossdifference of every pair of points on line X₂₄₈X₆₈₄

• X₁₃₂ = X₁₀₅-of-orthic triangle if ABC is acute

• X₁₃₂ = complement of X₁₂₉₇

• X₁₃₂ = perspector of circumconic centered at X₂₃₂

• X₁₃₂ = center of rectangular hyperbola {A, B, C,X₄,X₁₁₂,PU(39)}} that is locus of trilinear poles of lines passing through X₂₃₂

• X₁₃₂ = center of rectangular hyperbola {X₄,X₁₁₂,X₃₇₁,X₃₇₂,X₃₇₈,X₁₀₆₄}

• X₁₃₂ = inverse-in-polar-circle of X₉₈

• X₁₃₂ = circumcircle-inverse of X₃₄₁₃₁

• X₁₃₂ = inverse-in-{circumcircle, nine-point circle}-inverter of X₁₀₇

• X₁₃₂ = anticenter of cyclic quadrilateral ABCX₁₁₂

• X₁₃₂ = Λ(X₄, X₆), wrt orthic triangle

• X₁₃₂ = crosssum of circumcircle intercepts of line PU(37) (line X₃X₅₂₅)

• X₁₃₂ = orthopole of PU(37)