X₂₃₉

X₁-LINE CONJUGATE OF X₄₂

X₂₃₉ = X₁-line conjugate of X₄₂

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

X₂₃₉ (X₁-LINE CONJUGATE OF X₄₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 2}, {6, 75}

• Related Centers: X₁ (INCENTER), X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT), X₂ (CENTROID), X₆ (SYMMEDIAN POINT), X₇₅ (ISOTOMIC CONJUGATE OF INCENTER)

Trilinears: \(bc(a^{2} - bc) : :\)

Barycentrics: \(a^{2} - bc : :\)

Search Value (6, 9, 13): \(-17.268557204723\)

▸ Incident Central Lines21 lines

{1, 2}, {6, 75}, {7, 193}, {9, 192}, {44, 190}, {57, 330}, {63, 194}, {81, 274}, {83, 213}, {86, 1100}, {92, 607}, {141, 319}, {238, 740}, {241, 664}, {257, 333}, {294, 666}, {318, 458}, {320, 524}, {335, 518}, {514, 649}, {1043, 1104}

▸ Functional Relations20 relations

• X₂₃₉ = reflection of X_i in X_j for these (i,j): (190, 44), (320, 1086)

• X₂₃₉ = isogonal conjugate of X₂₉₂

• X₂₃₉ = isotomic conjugate of X₃₃₅

• X₂₃₉ = complement of X₆₅₄₂

• X₂₃₉ = anticomplement of X₃₉₁₂

• X₂₃₉ = crosspoint of X₂₅₆ and X₂₉₁

• X₂₃₉ = crosssum of X_i and X_j for these (i,j): (3, 255), (212, 219)

• X₂₃₉ = crossdifference of every pair of points on line X₄₂X₆₄₉

• X₂₃₉ = X_i-Hirst inverse of X_j for these (i,j): (171, 238), (665, 1015)

• X₂₃₉ = X₁-line conjugate of X₄₂

• X₂₃₉ = X_i-beth conjugate of X_j for these (i,j): (333, 239), (645, 44)

• X₂₃₉ = perspector of conic {A, B, C,X₈₆,X₁₉₀}

• X₂₃₉ = inverse-in-Steiner-circumellipse of X₁

• X₂₃₉ = trilinear pole of line X₆₅₉X₈₁₂

• X₂₃₉ = crossdifference of PU(8)

• X₂₃₉ = intersection of trilinear polars of PU(6) (the 1st and 2nd bicentrics of the Lemoine axis)

• X₂₃₉ = X₂-Ceva conjugate of X₆₆₅₁

• X₂₃₉ = trilinear pole of PU(134)

• X₂₃₉ = trilinear square root of X₃₉₀₄₄

• X₂₃₉ = barycentric square root of X₄₃₆₆

Notes

X₂₃₉ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(6) and U(6) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)

X₂₃₉ is the point of intersection of the following lines:

• X₁X₂ = trilinear polar of X₁₉₀

• trilinear polar of cevapoint <cevapoint>`{:ref:`X₁, X₂}, which is X₂₃₉X₅₁₄

• UV, where U = X₁-Ceva-conjugate-of-(2) = X₁₉₂, and V = X₂-Ceva-conjugate-of-X₁ = X₉ (Randy Hutson, December 26, 2015)