X₃₅₀

X₂-HIRST INVERSE OF X₇₅

X₃₅₀ is the X₂-hirst inverse of X₇₅.

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

X₃₅₀ (X₂-HIRST INVERSE OF X₇₅) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 37}, {1, 76}

• Related Centers: X₁₉₁₁ (TRILINEAR POLE OF PU(9)), X₂ (CENTROID), X₃₇ (CROSSPOINT OF INCENTER AND CENTROID), X₁ (INCENTER), X₇₆ (3rd BROCARD POINT)

Trilinears: \((a^{2} - bc)b^{2}c^{2} : (b^{2} - ca)c^{2}a^{2} : (c^{2} - ab)a^{2}b^{2}\)

Barycentrics: \(bc(a^{2} - bc) : ca(b^{2} - ca) : ab(c^{2} - ab)\)

Search Value (6, 9, 13): \(24.647406574100\)

▸ Incident Central Lines19 lines

{1, 76}, {2, 37}, {11, 325}, {33, 264}, {36, 99}, {42, 308}, {55, 183}, {69, 497}, {86, 310}, {172, 384}, {190, 672}, {256, 314}, {291, 726}, {305, 614}, {320, 513}, {447, 811}, {519, 668}, {538, 1015}, {889, 903}

▸ Functional Relations11 relations

• X₃₅₀ = isogonal conjugate of X₁₉₁₁

• X₃₅₀ = isotomic conjugate of X₂₉₁

• X₃₅₀ = complement of X₁₇₇₅₉

• X₃₅₀ = anticomplement of X₁₅₇₅

• X₃₅₀ = anticomplementary conjugate of X₂₀₃₃₅

• X₃₅₀ = crosspoint of X₂₅₇ and X₃₃₅

• X₃₅₀ = crossdifference of every pair of points on line X₂₁₃X₆₆₇

• X₃₅₀ = X₂-Hirst inverse of X₇₅

• X₃₅₀ = X₂-Ceva conjugate of X₃₉₀₂₈

• X₃₅₀ = perspector of hyperbola {A, B, C,X₂₇₄,X₆₆₈,X₈₇₄,PU(10)}}

• X₃₅₀ = intersection of trilinear polars of P(10) and U(10)

Notes

X₃₅₀ is the perspector of triangle ABC and the tangential triangle of the conic that passes through A, B, C, and the pair P(10) and U(10) of bicentric points (see the notes just before X₁₉₀₈. (Randy Hutson, 9/23/2011)