X₃₂₃

ISOTOMIC CONJUGATE OF X₉₄

X₃₂₃ is the isotomic conjugate of X₉₄.

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

X₃₂₃ (ISOTOMIC CONJUGATE OF X₉₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {23, 110}

• Related Centers: X₂₃ (FAR-OUT POINT), X₁₁₀ (FOCUS OF KIEPERT PARABOLA)

Trilinears: \(\sin 3A \csc^{2} A : \sin 3B \csc^{2} B : \sin 3C \csc^{2} C\)

Barycentrics: \(\sin 3A \csc A : \sin 3B \csc B : \sin 3C \csc C\)

Search Value (6, 9, 13): \(4.868692466079\)

▸ Alternate Coordinates3 forms

• Trilinears: \(4 \sin A - 3 \csc A :\)

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

• Barycentrics: directed distance of A to Hatzipolakis axis : :

▸ Incident Central Lines7 lines

{2, 6}, {20, 155}, {23, 110}, {30, 146}, {140, 195}, {187, 249}, {401, 525}

▸ Functional Relations18 relations

• X₃₂₃ = reflection of X₂₃ in X₁₁₀

• X₃₂₃ = isogonal conjugate of X₁₉₈₉

• X₃₂₃ = isotomic conjugate of X₉₄

• X₃₂₃ = complement of X₃₇₇₇₉

• X₃₂₃ = anticomplement of X₃₅₈₀

• X₃₂₃ = X₃₄₀-Ceva conjugate of X₁₈₆

• X₃₂₃ = cevapoint of X₆ and X₃₉₉

• X₃₂₃ = X₅₀-cross conjugate of X₁₈₆

• X₃₂₃ = crosssum of X₃₉₅ and X₃₉₆

• X₃₂₃ = crosssum of X₃₆₂₉₈ and X₃₆₂₉₉

• X₃₂₃ = crossdifference of every pair of points on line X₅₁X₅₁₂

• X₃₂₃ = crosspoint of X₆ and X₃₉₉ wrt both the excentral and tangential triangles

• X₃₂₃ = inverse-in-MacBeath-circumconic of X₂

• X₃₂₃ = orthocentroidal-to-ABC similarity image of X₁₁₀

• X₃₂₃ = 4th-Brocard-to-circumsymmedial similarity image of X₁₁₀

• X₃₂₃ = X₁₃₁₉₂-of-circumsymmedial-triangle

• X₃₂₃ = Vu circlecevian point V(X₁₅,X₁₆)

• X₃₂₃ = intersection of the tangent to hyperbola {A, B, C,X₆,X₁₃,X₁₆}} at X₁₆ and the tangent to hyperbola {A, B, C,X₆,X₁₄,X₁₅}} at X₁₅

Notes

Let A'B'C' be the Trinh triangle. Let A″ be the trilinear pole of line B'C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₂₃. (Randy Hutson, October 13, 2015)