X₃₂₁

ISOTOMIC CONJUGATE OF X₈₁

X₃₂₁ is the isotomic conjugate of X₈₁.

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

X₃₂₁ (ISOTOMIC CONJUGATE OF X₈₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {42, 1215}

• Related Centers: X₄₂ (CROSSPOINT OF INCENTER AND SYMMEDIAN POINT), X₁₂₁₅ (ISOGONAL CONJUGATE OF X₁₁₇₈)

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

Barycentrics: \(bc(b + c) : :\)

Search Value (6, 9, 13): \(4.172623472174\)

▸ Alternate Coordinates2 forms

• Trilinears: \(a(b + c)\csc(A - \omega ) : :\)

• Barycentrics: \(|AP(1)| + |AU(1)| : :\)

X₃₂₁ lies on the Kiepert hyperbola.

▸ Incident Central Lines16 lines

{1, 964}, {2, 37}, {4, 8}, {10, 756}, {38, 726}, {76, 561}, {81, 314}, {83, 213}, {98, 100}, {190, 333}, {226, 306}, {310, 335}, {313, 594}, {319, 1029}, {668, 671}, {693, 824}

▸ Functional Relations24 relations

• X₃₂₁ = reflection of X₄₂ in X₁₂₁₅

• X₃₂₁ = isogonal conjugate of X₁₃₃₃

• X₃₂₁ = isotomic conjugate of X₈₁

• X₃₂₁ = anticomplement of X₃₆₆₆

• X₃₂₁ = X_i-Ceva conjugate of X_j for (i,j) = (75,10), (76,313), (312,306)

• X₃₂₁ = cevapoint of X₃₇ and X₇₂

• X₃₂₁ = X₄₄₂-cross conjugate of X₂₆₄

• X₃₂₁ = crosspoint of X_i and X_j for these (i,j): (75, 76), (313, 349)

• X₃₂₁ = crosssum of X₃₁ and X₃₂

• X₃₂₁ = crossdifference of every pair of points on line X₆₆₇X₈₃₈

• X₃₂₁ = homothetic center of ABC and inverse of n(Medial)*n(Incentral) triangle

• X₃₂₁ = pole wrt polar circle of trilinear polar of X₂₈ (line X₅₁₃X₁₄₃₀)

• X₃₂₁ = polar conjugate of X₂₈

• X₃₂₁ = Danneels point of X₇₅

• X₃₂₁ = inverse-in-Fuhrmann-circle of X₅₀₁₆

• X₃₂₁ = barycentric product X₁₀₀*X₈₅₀

• X₃₂₁ = perspector of ABC and cross-triangle of Gemini triangles 13 and 14

• X₃₂₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 13

• X₃₂₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 14

• X₃₂₁ = perspector of ABC and cross-triangle of ABC and Gemini triangle 20

• X₃₂₁ = barycentric product of vertices of Gemini triangle 16

• X₃₂₁ = barycentric product of vertices of Gemini triangle 20

• X₃₂₁ = perspector of Gemini triangle 22 and cross-triangle of ABC and Gemini triangle 22

• X₃₂₁ = trilinear pole of line X₅₂₃X₁₅₇₇ (the perspectrix of ABC and Gemini triangles 21 and 27)

Notes

Let A₂₆B₂₆C₂₆ be Gemini triangle 26. Let A' be the perspector of conic {A, B, C,B₂₆,C₂₆}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₃₂₁. (Randy Hutson, January 15, 2019)