X₂₅₀

ISOGONAL CONJUGATE OF X₁₂₅

X₂₅₀ is the isogonal conjugate of X₁₂₅.

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

X₂₅₀ (ISOGONAL CONJUGATE OF X₁₂₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {23, 232}, {186, 249}

• Related Centers: X₁₂₅ (CENTER OF JERABEK HYPERBOLA), X₂₃ (FAR-OUT POINT), X₂₃₂ (-CEVA CONJUGATE OF X₁₃₂), X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄), X₂₄₉ (ISOGONAL CONJUGATE OF X₁₁₅)

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

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

Search Value (6, 9, 13): \(-0.110816929619\)

▸ Alternate Coordinates1 form

• Trilinears: \((a^{2}\sec A)/(b^{2} - c^{2})^{2} : (b^{2}\sec B)/(c^{2} - a^{2})^{2} : (c^{2}\sec C)/(a^{2} - b^{2})^{2}\)

▸ Incident Central Lines9 lines

{23, 232}, {107, 687}, {110, 520}, {112, 691}, {186, 249}, {325, 340}, {476, 933}, {523, 648}, {827, 935}

▸ Functional Relations8 relations

• X₂₅₀ = isogonal conjugate of X₁₂₅

• X₂₅₀ = isotomic conjugate of X₃₃₉

• X₂₅₀ = cevapoint of X_i and X_j for these (i,j): (3, 110), (25, 112), (162, 270), (1113, 1114)

• X₂₅₀ = X_i-cross conjugate of X_j for these (i,j): (3, 110), (22, 99), (24, 107), (25, 112), (199, 101)

• X₂₅₀ = polar conjugate of X₃₃₈

• X₂₅₀ = circumcenter of reflection triangle of X₁₂₅

• X₂₅₀ = barycentric product X₉₉*X₁₁₂

• X₂₅₀ = cevapoint of MacBeath circumconic intercepts of Brocard axis

Notes

X₂₅₀ is the trilinear pole of line X₁₁₀X₁₁₂, which is the tangent to the MacBeath circumconic at X₁₁₀, and the locus of trilinear poles of tangents at P to conic {A, B, C,X₃,P}}, as P moves on the Brocard axis. (Randy Hutson, March 21, 2019)