X₈₂₃

ISOGONAL CONJUGATE OF X₈₂₂

X₈₂₃ is the isogonal conjugate of X₈₂₂.

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

X₈₂₃ (ISOGONAL CONJUGATE OF X₈₂₂) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {100, 107}, {264, 379}

• Related Centers: X₈₂₂ (CROSSDIFFERENCE OF X₁ AND X₂₉), X₁₀₀ (ANTICOMPLEMENT OF FEUERBACH POINT), X₁₀₇ (Ψ(SYMMEDIAN POINT, ORTHOCENTER)), X₂₆₄ (ISOTOMIC CONJUGATE OF CIRCUMCENTER), X₃₇₉ (EULER LINE INTERCEPT OF LINE X)

Trilinears: \(1/(\sec^{2} B - \sec^{2} C) : :\)

Search Value (6, 9, 13): \(0.526843354611\)

▸ Alternate Coordinates3 forms

• Trilinears: \(1/(\csc B/2 - \csc C/2) : :\)

• Trilinears: \(1/(\csc 2B - \csc 2C) : :\)

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

▸ Incident Central Lines7 lines

{100, 107}, {110, 681}, {158, 897}, {264, 379}, {648, 651}, {662, 811}, {774, 821}

▸ Functional Relations9 relations

• X₈₂₃ = isogonal conjugate of X₈₂₂

• X₈₂₃ = isotomic conjugate of X₂₄₀₁₈

• X₈₂₃ = anticomplement of X₁₆₅₉₅

• X₈₂₃ = perspector of ABC and tangential triangle, wrt excentral triangle, of bianticevian conic of X₁ and X₄

• X₈₂₃ = trilinear pole of line X₁X₂₉

• X₈₂₃ = pole wrt polar circle of trilinear polar of X₆₅₆ (line X₂₆₃₁X₂₆₃₂, or PU(75))

• X₈₂₃ = X₄₈-isoconjugate (polar conjugate) of X₆₅₆

• X₈₂₃ = X₆-isoconjugate of X₅₂₀

• X₈₂₃ = trilinear product of Steiner circumellipse intercepts of van Aubel line

Notes

Let L_a be the A-extraversion of line X₁X₂₉, 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 X₈₂₃. (Randy Hutson, January 29, 2018)