X₂₄₉

ISOGONAL CONJUGATE OF X₁₁₅

X₂₄₉ is the isogonal conjugate of X₁₁₅.

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

X₂₄₉ (ISOGONAL CONJUGATE OF X₁₁₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {186, 250}, {187, 323}

• Related Centers: X₁₁₅ (CENTER OF KIEPERT HYPERBOLA), X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄), X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅), X₁₈₇ (INVERSE-IN-CIRCUMCIRCLE OF X), X₃₂₃ (ISOTOMIC CONJUGATE OF X₉₄)

Trilinears: \((\csc A)\csc^{2}(B - C) : (\csc B)\csc^{2}(C - A) : (\csc C)\csc^{2}(A - B)\)

Barycentrics: \(\csc^{2}(B - C) : \csc^{2}(C - A) : \csc^{2}(A - B)\)

Search Value (6, 9, 13): \(0.395641663632\)

▸ Alternate Coordinates1 form

• Trilinears: \(a/(b^{2} - c^{2})^{2} : b/(c^{2} - a^{2})^{2} : c/(a^{2} - b^{2})^{2}\)

▸ Incident Central Lines8 lines

{99, 525}, {110, 512}, {186, 250}, {187, 323}, {297, 316}, {648, 687}, {805, 827}, {849, 1110}

▸ Functional Relations10 relations

• X₂₄₉ = isogonal conjugate of X₁₁₅

• X₂₄₉ = isotomic conjugate of X₃₃₈

• X₂₄₉ = cevapoint of X_i and X_j for these (i,j): (6, 110), (24, 112), (1379, 1380)

• X₂₄₉ = X_i-cross conjugate of X_j for these (i,j): (3, 99), (6, 110)

• X₂₄₉ = perspector of ABC and reflection of symmedial triangle in the Brocard axis

• X₂₄₉ = polar conjugate of X₂₉₇₀

• X₂₄₉ = X₆₃-isoconjugate of X₈₇₅₄

• X₂₄₉ = X₉₂-isoconjugate of X₁₂₅

• X₂₄₉ = crossdifference of every pair of points on line X₁₆₄₈X₈₀₂₉

• X₂₄₉ = Vu circlecevian point of PU(2)

Notes

X₂₄₉ is the vertex conjugate of the foci of the inellipse that is the barycentric square of line X₂X₆. The center of this inellipse is X₆₂₀, and its perspector is X₄₅₉₀. (Randy Hutson, October 15, 2018)

X₂₄₉ is the trilinear pole of line X₁₁₀X₃₅₁, which is the tangent to the circumcircle at X₁₁₀, and the locus of trilinear poles of tangents at P to hyperbola {A, B, C,X₆,P}}, as P moves on the Brocard axis. (Randy Hutson, October 15, 2018)