X₁₀₈₆

CENTER OF HYPERBOLA {{A,B,C,X₂,X₇|}

X₁₀₈₆ = complement of X₁₉₀

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

X₁₀₈₆ (CENTER OF HYPERBOLA {{A,B,C,X₂,X₇|}) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {6, 7}, {2, 45}

• Related Centers: X₆ (SYMMEDIAN POINT), X₇ (GERGONNE POINT), X₂ (CENTROID), X₄₅ (-BETH CONJUGATE OF X₁)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = bc(b - c)^{2}\)

Barycentrics: \(g(a,b,c) : g(b,c,a) : g(c,a,b)\), where \(g(a,b,c) = (b - c)^{2}\)

Search Value (6, 9, 13): \(1.705536514047\)

X₁₀₈₆ lies on the Steiner inellipse.

▸ Incident Central Lines16 lines

{1, 528}, {2, 45}, {6, 7}, {8, 599}, {10, 537}, {11, 244}, {37, 142}, {44, 527}, {53, 273}, {57, 1020}, {75, 141}, {115, 116}, {220, 277}, {239, 320}, {812, 1015}, {918, 1111}

▸ Functional Relations17 relations

• X₁₀₈₆ = center of circumconic that is locus of trilinear poles of lines parallel to Gergonne line (i.e. lines that pass through X₅₁₄). This conic is the isotomic conjugate of the Nagel line. (Randy Hutson, September 14, 2016)

• X₁₀₈₆ = midpoint of X_i and X_j for these (i,j): (2, 903), (7, 673), (75, 335), (239, 320)

• X₁₀₈₆ = isogonal conjugate of X₁₂₅₂

• X₁₀₈₆ = isotomic conjugate of X₁₀₁₆

• X₁₀₈₆ = complement of X₁₉₀

• X₁₀₈₆ = crosspoint of X₂ and X₅₁₄

• X₁₀₈₆ = crosssum of X_i and X_j for these (i,j): (6, 101), (9, 1018), (32, 692), (219, 906)

• X₁₀₈₆ = crossdifference of every pair of points on line X₁₀₁X₆₉₂

• X₁₀₈₆ = perspector of circumconic centered at X₅₁₄

• X₁₀₈₆ = X₂-Ceva conjugate of X₅₁₄

• X₁₀₈₆ = projection from Steiner circumellipse to Steiner inellipse of X₉₀₃

• X₁₀₈₆ = trilinear pole of line X₇₆₄X₁₆₄₇

• X₁₀₈₆ = trilinear pole wrt medial triangle of Nagel line

• X₁₀₈₆ = anticomplement of X₄₄₂₂

• X₁₀₈₆ = barycentric product X₅₉₉₇*X₅₉₉₈

• X₁₀₈₆ = barycentric square of X₅₁₄

• X₁₀₈₆ = {X₃₆₆₁,X₃₆₆₂}-harmonic conjugate of X₁₇₂₂₇

Notes

The line f(a,b,c)x + f(b,c,a)y + f(c,a,b)z = 0 is tangent to the circumcircle at X₁₀₁. Also, X₁₀₈₆ is the point of tangency of the Steiner inscribed ellipse with the line tangent to the nine-point circle and the incircle. (Paul Yiu, #4197, 11/24/01).

Let A₇B₇C₇ and A₈B₈C₈ be the Gemini triangles 7 and 8. Let A' be the barycentric product A₇*A₈ and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₀₈₆. (Randy Hutson, November 30, 2018)