X₉₁₇

X₅₁₆-CROSS CONJUGATE OF X₄

X₉₁₇ = isogonal conjugate of X₉₁₆

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

X₉₁₇ (X₅₁₆-CROSS CONJUGATE OF X₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1305, 3}

• Related Centers: X₁₃₀₅ (COLLINGS TRANSFORM OF X₄₄₀), X₃ (CIRCUMCENTER)

Trilinears: \((\sec A)/((a - b) \sin 2B + (a - c) \sin 2C) : :\)

Barycentrics: \(1/((a^{2} - b^{2} - c^{2}) (a^{3} (b^{2} + c^{2}) - a^{2} (b^{3} + c^{3}) - a (b^{2} - c^{2})^{2} + b^{5} - b^{3} c^{2} - b^{2} c^{3} + c^{5})) : :\)

Search Value (6, 9, 13): \(14.158875247015\)

X₉₁₇ lies on the circumcircle.

▸ Incident Central Lines4 lines

{4, 101}, {27, 110}, {92, 100}, {109, 278}

▸ Functional Relations9 relations

• X₉₁₇ = reflection of X₁₃₀₅ in X₃

• X₉₁₇ = reflection of X₄ in X₅₁₉₀

• X₉₁₇ = isogonal conjugate of X₉₁₆

• X₉₁₇ = X₅₁₆-cross conjugate of X₄

• X₉₁₇ = polar conjugate of isotomic conjugate of X₂₉₈₉

• X₉₁₇ = X₆₃-isoconjugate of X₈₆₀₈

• X₉₁₇ = antipode of X₄ in hyperbola {A, B, C,X₄,X₂₇}

• X₉₁₇ = polar-circle-inverse of X₁₁₈

• X₉₁₇ = the point of intersection, other than A, B, and C, of the circumcircle and hyperbola {A, B, C,X₄,X₂₇}

Notes

Let A'B'C' be the 1st circumperp triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X₉₁₇. (Randy Hutson, July 11, 2019)