X₃₃₁

ISOTOMIC CONJUGATE OF X₂₁₉

X₃₃₁ is the isotomic conjugate of X₂₁₉.

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

X₃₃₁ (ISOTOMIC CONJUGATE OF X₂₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {4, 150}, {85, 92}

• Related Centers: X₂₁₉ (-CEVA CONJUGATE OF X₅₅), X₄ (ORTHOCENTER), X₁₅₀ (REFLECTION OF X₂₀ IN X₁₀₃), X₈₅ (ISOTOMIC CONJUGATE OF X₉), X₉₂ (CEVAPOINT OF INCENTER AND CLAWSON POINT)

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

Barycentrics: \(\sec^{2}(A/2) \sec A : \sec^{2}(B/2) \sec B : \sec^{2}(C/2) \sec C\)

Search Value (6, 9, 13): \(-1.648176523381\)

▸ Alternate Coordinates1 form

• Trilinears: \((1 - \sec A)\csc(A - \omega ) : (1 - \sec B)\csc(B - \omega ) : (1 - \sec C)\csc(C - \omega )\)

▸ Incident Central Lines7 lines

{4, 150}, {7, 286}, {34, 870}, {75, 225}, {85, 92}, {108, 767}, {274, 278}

▸ Functional Relations8 relations

• X₃₃₁ = isogonal conjugate of polar conjugate of isotomic conjugate of X₆₀₅₆

• X₃₃₁ = isotomic conjugate of X₂₁₉

• X₃₃₁ = cevapoint of X_i and X_j for these (i,j): (7, 278), (92, 273)

• X₃₃₁ = X₉₂-cross conjugate of X₂₆₄

• X₃₃₁ = X₁₉-isoconjugate of X₆₀₅₆

• X₃₃₁ = trilinear pole of polar of X₅₅ wrt polar circle

• X₃₃₁ = pole wrt polar circle of trilinear polar of X₅₅ (line X₆₅₇X₆₆₃)

• X₃₃₁ = polar conjugate of X₅₅

Notes

X₃₃₁ is the Brianchon point (perspector) of the inconic that is the polar conjugate of the isogonal conjugate of the incircle. (Randy Hutson, October 15, 2018)