X₁₃₀₉

COLLINGS TRANSFORM OF X₈₆₀

X₁₃₀₉ = Λ(trilinear polar of X₉₀₅)

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

X₁₃₀₉ (COLLINGS TRANSFORM OF X₈₆₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {105, 243}, {102, 515}

• Related Centers: X₁₀₅ (Λ(INCENTER, SYMMEDIAN POINT)), X₂₄₃ (-LINE CONJUGATE OF X₇₃), X₁₀₂ (Λ(INCENTER, ORTHOCENTER)), X₅₁₅ (ISOGONAL CONJUGATE OF X₁₀₂)

Trilinears: \((\csc 2A)/[4(\sin^{2} B - \sin^{2} C) \cos B \cos C + \sin^{2} B \cos B - \sin^{2} C \cos C] : :\)

Search Value (6, 9, 13): \(1.541677982293\)

▸ Alternate Coordinates1 form

• Trilinears: \(bc/[(b - c)(a^{2} - b^{2} - c^{2})(a^{2}b + a^{2}c - 2abc + b^{2}c + bc^{2} - b^{3} - c^{3})] : :\)

X₁₃₀₉ lies on the circumcircle.

▸ Incident Central Lines5 lines

{4, 953}, {102, 515}, {105, 243}, {109, 522}, {693, 934}

▸ Functional Relations14 relations

• X₁₃₀₉ = isogonal conjugate of X₈₆₇₇

• X₁₃₀₉ = anticomplement of X₁₀₀₁₇

• X₁₃₀₉ = cevapoint of X_i and X_j for these (i,j): (515, 522), (523, 860)

• X₁₃₀₉ = Λ(X₁₄₅₉, X₁₉₄₆)

• X₁₃₀₉ = Λ(trilinear polar of X₉₀₅)

• X₁₃₀₉ = trilinear pole of line X₆X₂₈₁

• X₁₃₀₉ = circumcircle-antipode of X₂₇₃₄

• X₁₃₀₉ = Ψ(X₃, X₈)

• X₁₃₀₉ = Ψ(X₆, X₂₈₁)

• X₁₃₀₉ = inverse-in-polar-circle of X₃₂₅₉

• X₁₃₀₉ = pole wrt polar circle of Sherman line (line X₃₂₅₉X₃₃₂₆) (see http://forumgeom.fau.edu/FG2012volume12/FG201220.pdf)

• X₁₃₀₉ = intersection of antipedal lines of X₁₀₂ and X₁₀₉

• X₁₃₀₉ = polar conjugate of X₁₀₀₁₅

• X₁₃₀₉ = X₆₃-isoconjugate of X₃₃₁₀

Notes

Let A', B', C' be the intersections of line X₄X₈ and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₁₃₀₉. (Randy Hutson, January 29, 2018)