X₈₅

ISOTOMIC CONJUGATE OF X₉

X₈₅ is the isotomic conjugate of X₉.

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

X₈₅ (ISOTOMIC CONJUGATE OF X₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {7, 8}, {29, 34}

• Related Centers: X₉ (MITTENPUNKT), X₇ (GERGONNE POINT), X₈ (NAGEL POINT), X₂₉ (CEVAPOINT OF INCENTER AND ORTHOCENTER), X₃₄ (-BETH CONJUGATE OF X₄)

Trilinears: \(b^{2}c^{2}/(b + c - a) : c^{2}a^{2}/(c + a - b) : a^{2}b^{2}/(a + b - c)\)

Barycentrics: \(bc/(b + c - a) : ca/(c + a - b) : ab/(a + b - c)\)

Search Value (6, 9, 13): \(1.369706806417\)

▸ Alternate Coordinates2 forms

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

• Barycentrics: \(\cot A' : :\), where J_aJ_bJ_c is the excentral triangle

▸ Incident Central Lines12 lines

{1, 664}, {2, 241}, {7, 8}, {12, 120}, {29, 34}, {56, 870}, {57, 274}, {76, 226}, {92, 331}, {109, 767}, {150, 355}, {264, 309}

▸ Functional Relations15 relations

• X₈₅ = isogonal conjugate of X₄₁

• X₈₅ = isotomic conjugate of X₉

• X₈₅ = complement of X₃₁₇₇

• X₈₅ = anticomplement of X₁₂₁₂

• X₈₅ = X₂₇₄-Ceva conjugate of X₃₄₈

• X₈₅ = cevapoint of X_i and X_j for these (i,j): (1, 169), (2, 7), (57, 77), (92, 342)

• X₈₅ = X_i-cross conjugate of X_j for these (i,j): (2, 75), (57, 273), (92, 309), (142, 2), (226, 7)

• X₈₅ = X_i-beth conjugate of X_j for these (i,j): (76, 76), (85, 279), (99, 1), (274, 85), (668, 85), (789, 85), (799, 85), (811, 85)

• X₈₅ = trilinear pole of line X₅₂₂X₆₉₃ (the isotomic conjugate of the circumconic centered at X₁, conic {A, B, C,X₁₀₀,X₆₆₄,X₁₁₂₀,X₁₃₂₀}; also the polar of X₃₃ wrt polar circle)

• X₈₅ = pole wrt polar circle of trilinear polar of X₃₃ (line X₆₅₇X₄₀₄₁)

• X₈₅ = polar conjugate of X₃₃

• X₈₅ = {X₇,X₈}-harmonic conjugate of X₆₆₀₄

• X₈₅ = trilinear square of X₅₀₈

• X₈₅ = trilinear product of vertices of Gemini triangle 9

• X₈₅ = trilinear product of vertices of Gemini triangle 10

Notes

Let A₃₈B₃₈C₃₈ be Gemini triangle 38. Let A' be the perspector of conic {A, B, C,B₃₈,C₃₈}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₅. (Randy Hutson, January 15, 2019)