X₈₁₁

ISOGONAL CONJUGATE OF X₈₁₀

X₈₁₁ is the isogonal conjugate of X₈₁₀.

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

X₈₁₁ (ISOGONAL CONJUGATE OF X₈₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {99, 108}, {1, 336}

• Related Centers: X₈₁₀ (CROSSPOINT OF X₁ AND X₁₆₃), X₉₉ (STEINER POINT), X₁₀₈ (Ψ(CIRCUMCENTER, INCENTER)), X₁ (INCENTER), X₃₃₆ (ISOTOMIC CONJUGATE OF X₂₄₀)

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

Barycentrics: \((\sec A)/(\cos^{2} B - \cos^{2} C) : :\)

Search Value (6, 9, 13): \(-2.269529969762\)

▸ Alternate Coordinates4 forms

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

• Barycentrics: \((\sec A)/(\sin^{2} B - \sin^{2} C) : :\)

• Barycentrics: \(\csc 2A \csc(B - C) : :\)

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

▸ Incident Central Lines8 lines

{1, 336}, {75, 1099}, {99, 108}, {112, 789}, {162, 799}, {350, 447}, {645, 648}, {662, 823}

▸ Functional Relations9 relations

• X₈₁₁ = isogonal conjugate of X₈₁₀

• X₈₁₁ = isotomic conjugate of X₆₅₆

• X₈₁₁ = complement of anticomplementary conjugate of X₂₁₃₀₀

• X₈₁₁ = anticomplement of X₁₆₅₇₃

• X₈₁₁ = trilinear pole of line X₁₉X₂₇

• X₈₁₁ = polar conjugate of X₆₆₁

• X₈₁₁ = perspector of ABC and tangential triangle, wrt anticevian triangle of X₁₉, of bianticevian conic of X₁ and X₄

• X₈₁₁ = trilinear product X₄*X₉₉

• X₈₁₁ = trilinear product X₂₄₇₉*X₂₄₈₀

Notes

Let L_a be the A-extraversion of line X₁₉X₂₇, and define L_b, L_c cyclically. Let A' = L_b ∩ L_c, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₁₁. (Randy Hutson, January 29, 2018)

As the trilinear product of Steiner circumellipse antipodes, X₈₁₁ lies on conic {A, B, C,X₆₆₈,X₇₈₉}} with center X₆₃₇₆ and perspector X₇₅. (Randy Hutson, July 11, 2019)