X₆₄₃–X₆₉₃

Crossdifferences & Trilinear Poles

Trilinear multipliers for the Kiepert and Yff parabolas, crossdifferences of classical centers, and trilinear poles of central lines.

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

X₆₄₃–X₆₉₃ (Crossdifferences & Trilinear Poles) — 16 representative centers (from 51 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

X₆₄₃TRILINEAR MULTIPLIER FOR KIEPERT PARABOLA

X₆₄₃ satisfies the equation X*(incircle) = Kiepert parabola, where * denotes trilinear multiplication, defined by

X₆₄₄TRILINEAR MULTIPLIER FOR YFF PARABOLA

X₆₄₄ satisfies the equation X*(incircle) = Yff parabola, where * denotes trilinear multiplication, defined by

X₆₄₅BARYCENTRIC MULTIPLIER FOR KIEPERT PARABOLA

X₆₄₅ satisfies the equation X*(incircle) = Kiepert parabola, where * denotes barycentric multiplication, defined by

X₆₄₆BARYCENTRIC MULTIPLIER FOR YFF PARABOLA

X₆₄₆ satisfies the equation X*(incircle) = Yff parabola, where * denotes barycentric multiplication, defined by

X₆₄₇CROSSDIFFERENCE OF X₂ AND X₃

X₆₄₇ is the point whose trilinears are coefficients for the Euler line.

X₆₄₈TRILINEAR POLE OF EULER LINE

X₆₄₈ is constructed as the pole of the Euler line L as follows: let A″, B″, C″ be the points where L meets the sidelines BC, CA, AB of ABC. Let A', B', C' be the harmonic conjugates of A″, B″, C″ with respect to {B,C}, {C,A}, {A,B}, respectively, The lines AA', BB', CC' concur in X₆₄₈.

X₆₄₉CROSSDIFFERENCE OF X₁ and X₂

X₆₄₉ = crossdifference of every pair of points on line {1, 2}

X₆₅₀CROSSDIFFERENCE OF X₁ AND X₃

X₆₅₀ = crossdifference of every pair of points on line {1, 3}

X₆₅₁TRILINEAR POLE OF LINE X₁X_3

X₆₅₁ = cevapoint of X₁₀₁ and X₁₀₉

X₆₅₂CROSSDIFFERENCE OF X₁ AND X₄

X₆₅₂ = crossdifference of every pair of points on line X₁X₄

X₆₅₃TRILINEAR POLE OF LINE X₁X_4

X₆₅₃ = polar conjugate of X₅₂₂

X₆₅₄CROSSDIFFERENCE OF X₁ AND X₅

X₆₅₄ = reflection of X₆₅₄ in the anti-orthic axis

X₆₅₅TRILINEAR POLE OF LINE X₁X_5

X₆₅₅ = trilinear pole of line {1, 5}

X₆₅₆CROSSDIFFERENCE OF X₁ AND X₁₉

X₆₅₆ = crossdifference of every pair of points on line {1, 19}

X₆₅₇CROSSDIFFERENCE OF X₁ AND X₇

X₆₅₇ = crossdifference of every pair of points on line {1, 7}

X₆₅₈TRILINEAR POLE OF LINE X₁X_7

X₆₅₈ = trilinear pole of line {1, 7}

X₆₅₉CROSSDIFFERENCE OF X₁ AND X₃₉

X₆₅₉ = X₉₈-Ceva conjugate of X₁₁

X₆₆₀TRILINEAR POLE OF LINE X₁X_39

X₆₆₀ = trilinear pole of line {1, 39}

X₆₆₁CROSSDIFFERENCE OF X₁ AND X₂₁

X₆₆₁ = barycentric product of PU(71)

X₆₆₂TRILINEAR POLE OF LINE X₁X_21

Let A₂B₂C₂ and A₃B₃C₃ be the 2nd and 3rd Parry triangles. Let A' be the trilinear product A₂*A₃, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₆₂. (Randy Hutson, February 10, 2016)

X₆₆₃CROSSDIFFERENCE OF X₂ AND X₇

X₆₆₃ = polar conjugate of the isotomic conjugate of X₆₅₂

X₆₆₄TRILINEAR POLE OF LINE X₂X_7

X₆₆₄ = trilinear pole of line {2, 7}

X₆₆₅CROSSDIFFERENCE OF X₂ AND X₁₁

X₆₆₅ = crossdifference of every pair of points on line {2, 11}

X₆₆₆TRILINEAR POLE OF LINE X₂X_11

X₆₆₆ = trilinear pole of line {2, 11}

X₆₆₇CROSSDIFFERENCE OF X₂ AND X₃₇

X₆₆₇ = reflection of X₆₆₇ in the Lemoine axis

X₆₆₈TRILINEAR POLE OF LINE X₂X_37

As the trilinear product of Steiner circumellipse antipodes, X₆₆₈ lies on conic {A, B, C,X₆₆₈,X₇₈₉}} with center X₆₃₇₆ and perspector X₇₅. (Randy Hutson, July 11, 2019)

X₆₆₉CROSSDIFFERENCE OF X₂ AND X₃₉

X₆₆₉ = crossdifference of every pair of points on line {2, 39}

X₆₇₀TRILINEAR POLE OF LINE X₂X_39

X₆₇₀ = trilinear pole of line {2, 39}

X₆₇₁TRILINEAR POLE OF LINE X₂X_523

X₆₇₁ = polar conjugate of X₄₆₈

X₆₇₂CROSSDIFFERENCE OF X₁ AND X₅₁₄

X₆₇₂ = X₂₃₂-of-excentral-triangle

X₆₇₃TRILINEAR POLE OF LINE X₁X_514

X₆₇₃ = BSS(a^2→a) of X₉₈

X₆₇₄CROSSDIFFERENCE OF X₆ AND X₅₁₄

As the isogonal conjugate of a point on the circumcircle, X₆₇₄ lies on the line at infinity.

X₆₇₅TRILINEAR POLE OF LINE X₆X_514

X₆₇₅ = inverse-in-orthoptic-circle-of-Steiner-inellipse of X₁₁₆

X₆₇₆CROSSDIFFERENCE OF X₃ AND X₁₀₁

X₆₇₆ = crosspoint of X₁₀₅ and X₁₀₈

X₆₇₇TRILINEAR POLE OF LINE X₃X_101

X₆₇₇ = isogonal conjugate of X₆₇₆

X₆₇₈CROSSPOINT OF X₁ AND X₄₄

X₆₇₈ is the crosspoint of X₁ and X₄₄.

X₆₇₉ISOGONAL CONJUGATE OF X₆₇₈

X₆₇₉ is the isogonal conjugate of X₆₇₈.

X₆₈₀CROSSDIFFERENCE OF X₆ AND X₁₅₈

As the isogonal conjugate of a point on the circumcircle, X₆₈₀ lies on the line at infinity.

X₆₈₁TRILINEAR POLE OF LINE X₆X_158

X₆₈₁ = isogonal conjugate of X₆₈₀

X₆₈₂POINT ARNEB

X₆₈₂ = crosspoint of X₃ and X₃₂

X₆₈₃ISOGONAL CONJUGATE OF X₆₈₂

X₆₈₃ is the isogonal conjugate of X₆₈₂.

X₆₈₄CROSSDIFFERENCE OF X₄ AND X₃₂

X₆₈₄ = perspector of hyperbola {A, B, C,X₃,X₅₁₁}}

X₆₈₅TRILINEAR POLE OF LINE X₄X_98

X₆₈₅ = trilinear pole of line X₄X₃₂

X₆₈₆CROSSDIFFERENCE OF X₄ AND X₁₁₀

X₆₈₆ = perspector of hyperbola {A, B, C,X₃,X₅₂₃}}

X₆₈₇TRILINEAR POLE OF LINE X₄X_110

X₆₈₇ = isogonal conjugate of X₆₈₆

X₆₈₈CROSSDIFFERENCE OF X₆ AND X₇₆

As the isogonal conjugate of a point on the circumcircle, X₆₈₈ lies on the line at infinity.

X₆₈₉TRILINEAR POLE OF LINE X₆X_76

X₆₈₉ = Ψ(X₃₂, X₂)

X₆₉₀CROSSDIFFERENCE OF LINE X₆ AND X₁₁₀

As the isogonal conjugate of a point on the circumcircle, X₆₉₀ lies on the line at infinity.

X₆₉₁TRILINEAR POLE OF LINE X₆X_110

X₆₉₁ = cevapoint of X₆ and X₃₅₁

X₆₉₂-CEVA CONJUGATE OF X₁₀₁

X₆₉₂ = isogonal conjugate of X₆₉₃

X₆₉₃ISOTOMIC CONJUGATE OF X₁₀₀

X₆₉₃ is the isotomic conjugate of X₁₀₀.