X₂₉₂

X₁-HIRST INVERSE OF X₂₉₁

X₂₉₂ is the X₁-hirst inverse of X₂₉₁.

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

X₂₉₂ (X₁-HIRST INVERSE OF X₂₉₁) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {1, 39}, {9, 87}

• Related Centers: X₂₃₉ (-LINE CONJUGATE OF X₄₂), X₁ (INCENTER), X₃₉ (BROCARD MIDPOINT), X₉ (MITTENPUNKT), X₈₇ (-CROSS CONJUGATE OF X₁)

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

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

Search Value (6, 9, 13): \(-0.125095805476\)

▸ Incident Central Lines11 lines

{1, 39}, {2, 334}, {6, 869}, {9, 87}, {37, 86}, {44, 660}, {58, 101}, {106, 813}, {171, 893}, {269, 1020}, {659, 665}

▸ Functional Relations15 relations

• X₂₉₂ = isogonal conjugate of X₂₃₉

• X₂₉₂ = isotomic conjugate of X₁₉₂₁

• X₂₉₂ = complement of X₂₀₃₄₅

• X₂₉₂ = anticomplement of X₂₀₃₃₃

• X₂₉₂ = X_i-Ceva conjugate of X_j for these (i,j): (2, 9470), (335, 292), (813, 3572)

• X₂₉₂ = cevapoint of X₁₇₁ and X₂₃₈

• X₂₉₂ = crossdifference of every pair of points on line X₆₅₉X₈₁₂

• X₂₉₂ = crossdifference of PU(134)

• X₂₉₂ = X₁-Hirst inverse of X₂₉₁

• X₂₉₂ = trilinear pole of line X₄₂X₆₄₉

• X₂₉₂ = point of intersection, other than A, B, C, of 1st and 2nd bicentrics of the Steiner circumellipse

• X₂₉₂ = perspector of hyperbola {A, B, C,X₆₆₀,X₈₁₃}}

• X₂₉₂ = perspector of ABC and unary cofactor triangle of obverse triangle of X₁

• X₂₉₂ = perspector of ABC and unary cofactor triangle of trilinear obverse triangle of X₂

• X₂₉₂ = trilinear square root of isogonal conjugate of X₃₉₀₄₄

Notes

Let A₃₃B₃₃C₃₃ be Gemini triangle 33. Let A' be the perspector of conic {A, B, C,B₃₃,C₃₃}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₂₉₂. (Randy Hutson, January 15, 2019)