X₃₁₈

ISOTOMIC CONJUGATE OF X₇₇

X₃₁₈ is the isotomic conjugate of X₇₇.

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

X₃₁₈ (ISOTOMIC CONJUGATE OF X₇₇) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 8}, {29, 33}

• Related Centers: X₇₇ (ISOGONAL CONJUGATE OF X₃₃), X₄ (ORTHOCENTER), X₈ (NAGEL POINT), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₃₃ (PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES)

Trilinears: \((1 + \sec A)/a^{2} : :\)

Barycentrics: \((1 + \sec A)/a : (1 + \sec B)/b : (1 + \sec C)/c\)

Search Value (6, 9, 13): \(5.323830333536\)

▸ Alternate Coordinates2 forms

• Trilinears: \(\sec A \csc^{2} A/2 : :\)

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

▸ Incident Central Lines17 lines

{2, 280}, {4, 8}, {10, 158}, {29, 33}, {53, 594}, {63, 412}, {75, 225}, {77, 309}, {108, 404}, {200, 1089}, {208, 653}, {239, 458}, {243, 958}, {253, 342}, {281, 346}, {317, 319}, {475, 1068}

▸ Functional Relations9 relations

• X₃₁₈ = isogonal conjugate of X₆₀₃

• X₃₁₈ = isotomic conjugate of X₇₇

• X₃₁₈ = X₂₆₄-Ceva conjugate of X₉₂

• X₃₁₈ = cevapoint of X₉ and X₃₃

• X₃₁₈ = X_i-cross conjugate of X_j for these (i,j): (9, 312), (10, 8), (281, 92)

• X₃₁₈ = trilinear pole of line X₃₀₆₄X₃₂₃₉ (the radical axis of Mandart circle and excircles radical circle, and the polar of X₅₇ wrt polar circle)

• X₃₁₈ = pole wrt polar circle of trilinear polar of X₅₇ (line X₅₁₃X₆₆₃)

• X₃₁₈ = polar conjugate of X₅₇

• X₃₁₈ = inverse-in-Fuhrmann-circle of X₅₀₈₁

Notes

Let A'B'C' be the mixtilinear incentral triangle. Let A″ be the pole, wrt the polar circle, of line B'C', and define B″, C″ cyclically. The lines AA″, BB″, CC″ concur in X₃₁₈. (Randy Hutson, November 30, 2018)