X₁₂₉₆

COLLINGS TRANSFORM OF X₁₂₆

X₁₂₉₆ = cevapoint of X₅₁₂ and X₅₇₄

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

X₁₂₉₆ (COLLINGS TRANSFORM OF X₁₂₆) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 111}, {4, 126}

• Related Centers: X₁₄₉₉ (POINT BIHAM), X₃ (CIRCUMCENTER), X₁₁₁ (PARRY POINT), X₄ (ORTHOCENTER), X₁₂₆ (-OF-MEDIAL-TRIANGLE)

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\) (see note above X₁₂₈₆)

Barycentrics: \(a^{2}/((b^{2} - c^{2}) (5 a^{2} - b^{2} - c^{2})) : :\)

Search Value (6, 9, 13): \(12.357283911768\)

X₁₂₉₆ lies on the circumcircle.

▸ Incident Central Lines5 lines

{3, 111}, {4, 126}, {98, 376}, {182, 729}, {511, 843}

▸ Functional Relations22 relations

• X₁₂₉₆ = reflection of X_i in X_j for these (i,j): (4, 126), (111, 3)

• X₁₂₉₆ = isogonal conjugate of X₁₄₉₉

• X₁₂₉₆ = cevapoint of X₅₁₂ and X₅₇₄

• X₁₂₉₆ = trilinear pole of line X₆X₃₇₃

• X₁₂₉₆ = Ψ(X₆, X₃₇₃)

• X₁₂₉₆ = reflection of X₂₆₉₆ in the Euler line

• X₁₂₉₆ = reflection of X₂₇₀₉ in the Brocard axis

• X₁₂₉₆ = reflection of X₂₇₄₆ in line X₁X₃

• X₁₂₉₆ = reflection of X₇₄ in line X₃X₃₅₁

• X₁₂₉₆ = X₁₃₈-of-hexyl-triangle

• X₁₂₉₆ = X₇₄-of-circumsymmedial-triangle

• X₁₂₉₆ = Λ(trilinear polar of X₁₉₉₂)

• X₁₂₉₆ = 1st-Parry-to-ABC similarity image of X₁₁₁

• X₁₂₉₆ = X₄ of 4th anti-Brocard triangle

• X₁₂₉₆ = perspector of 4th anti-Brocard and 1st Ehrmann triangles

• X₁₂₉₆ = Cundy-Parry Phi transform of X₁₄₂₆₃

• X₁₂₉₆ = Cundy-Parry Psi transform of X₃₄₁₆₁

• X₁₂₉₆ = Thomson-isogonal conjugate of X₅₂₄

• X₁₂₉₆ = Lucas-isogonal conjugate of X₅₂₄

• X₁₂₉₆ = perspector of ABC and 1st anti-Parry triangle

• X₁₂₉₆ = X₁₁₁-of-1st-anti-Parry-triangle

• X₁₂₉₆ = X₉₁₅₆-of-2nd-anti-Parry-triangle

Notes

Let L_A be the reflection of the line X₂X₆ in the line BC, and define L_B and L_C cyclically. Let A' = L_B∩L_C, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₁₂₉₆. (Randy Hutson, 9/23/2011)