X₈₈

ISOGONAL CONJUGATE OF X₄₄

X₈₈ is the isogonal conjugate of X₄₄.

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

X₈₈ (ISOGONAL CONJUGATE OF X₄₄) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 45}, {6, 89}

• Related Centers: X₄₄ (-LINE CONJUGATE OF X₁), X₂ (CENTROID), X₄₅ (-BETH CONJUGATE OF X₁), X₆ (SYMMEDIAN POINT), X₈₉ (ISOGONAL CONJUGATE OF X₄₅)

Trilinears: \(1/(b + c - 2a) : 1/(c + a - 2b) : 1/(a + b - 2c)\)

Barycentrics: \(a/(b + c - 2a) : b/(c + a - 2b) : c/(a + b - 2c)\)

Search Value (6, 9, 13): \(0.562219245046\)

▸ Linear Combinations1 combination

• X₈₈ = \((3r^{2} + 6rR - s^{2})\) X₁ + \(9rR\) X₂ - \(6r^{2}\) X₃ (Peter Moses, April 2, 2013)

▸ Incident Central Lines12 lines

{1, 100}, {2, 45}, {6, 89}, {28, 162}, {44, 679}, {57, 651}, {81, 662}, {105, 901}, {274, 799}, {278, 653}, {279, 658}, {291, 660}

▸ Functional Relations17 relations

• X₈₈ = isogonal conjugate of X₄₄

• X₈₈ = isotomic conjugate of X₄₃₅₈

• X₈₈ = complement of X₃₀₅₇₈

• X₈₈ = cevapoint of X_i and X_j for these (i,j): (1, 44), (6, 36)

• X₈₈ = X_i-cross conjugate of X_j for these (i,j): (44, 1), (517, 7)

• X₈₈ = X_i-aleph conjugate of X_j for these (i,j): (88, 1), (679, 88), (903, 63), (1022, 1052)

• X₈₈ = X₃₃₃-beth conjugate of X₁₉₀

• X₈₈ = trilinear product of PU(50)

• X₈₈ = perspector of conic {A, B, C,PU(50)}}

• X₈₈ = trilinear pole of PU(55); the line X₁X₅₁₃, the line through X₁ parallel to its trilinear polar; also normal to Feuerbach hyperbola at X₁

• X₈₈ = crossdifference of every pair of points on line X₆₇₈X₁₆₃₅

• X₈₈ = X₆-isoconjugate of X₅₁₉

• X₈₈ = BSS(a^2→a) of X₁₁₁

• X₈₈ = polar conjugate of X₃₈₄₆₂

• X₈₈ = X₁₉-isoconjugate of X₅₄₄₀

• X₈₈ = X₄₈-isoconjugate of X₃₈₄₆₂

• X₈₈ = X₉₂-isoconjugate of X₂₃₂₀₂

Notes

Let A₉B₉C₉ be Gemini triangle 9. Let A' be the perspector of conic {A, B, C,B₉,C₉}}, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₈₈. (Randy Hutson, January 15, 2019)

See (and hear) Dan Reznik’s Dance of the Swans: X₈₈ and X₁₆₂ and their Never-Touching Motion Over the Elliptic Billiard (March 4, 2020)