X₉₃

ISOGONAL CONJUGATE OF X₄₉

X₉₃ is the isogonal conjugate of X₄₉.

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

X₉₃ (ISOGONAL CONJUGATE OF X₄₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {49, 94}, {186, 252}

• Related Centers: X₄₉ (CENTER OF SINE-TRIPLE-ANGLE CIRCLE), X₉₄ (ISOGONAL CONJUGATE OF X₅₀), X₁₈₆ (INVERSE-IN-CIRCUMCIRCLE OF X₄), X₂₅₂ (ISOGONAL CONJUGATE OF X₁₄₃)

Trilinears: \(\sec 3A : \sec 3B : \sec 3C\)

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

Search Value (6, 9, 13): \(18.573417905067\)

▸ Alternate Coordinates1 form

• Barycentrics: \(1/(a^{2} (a^{2} - b^{2} - c^{2}) (a^{4} + b^{4} + c^{4} - 2 a^{2} b^{2} - 2 a^{2} c^{2} - b^{2} c^{2})) : :\)

▸ Incident Central Lines3 lines

{4, 562}, {49, 94}, {186, 252}

▸ Functional Relations5 relations

• X₉₃ = isogonal conjugate of X₄₉

• X₉₃ = anticomplement of X₃₄₈₃₃

• X₉₃ = X₅₀-cross conjugate of X₉₄

• X₉₃ = polar conjugate of X₁₉₉₄

• X₉₃ = X₂₉₆₄-isoconjugate of X₃

Notes

Let O_AO_BO_C be the Kosnita triangle. Let A' be the pole wrt polar circle of line O_BO_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₉₃. (Randy Hutson, June 7, 2019)