X₆₉₁

TRILINEAR POLE OF LINE X₆X_110

X₆₉₁ = cevapoint of X₆ and X₃₅₁

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

X₆₉₁ (TRILINEAR POLE OF LINE X₆X_110) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {30, 98}, {23, 111}

• Related Centers: X₆₉₀ (CROSSDIFFERENCE OF LINE X), X₃₀ (EULER INFINITY POINT), X₉₈ (TARRY POINT), X₂₃ (FAR-OUT POINT), X₁₁₁ (PARRY POINT)

Trilinears: \(a/[(b^{2} - c^{2})(2a^{2} - b^{2} - c^{2})] : :\)

Search Value (6, 9, 13): \(-0.628200957308\)

X₆₉₁ lies on the circumcircle, the circle {X₂,X₃,X₆,X₁₁₁}.

▸ Incident Central Lines14 lines

{3, 842}, {6, 843}, {23, 111}, {30, 98}, {74, 511}, {99, 523}, {110, 249}, {112, 250}, {182, 2698}, {316, 858}, {376, 477}, {741, 923}, {759, 897}, {805, 882}

▸ Functional Relations23 relations

• X₆₉₁ = reflection of X_i in X_j for these (i,j): (23, 187), (316, 858), (842, 3)

• X₆₉₁ = isogonal conjugate of X₆₉₀

• X₆₉₁ = isotomic conjugate of X₃₅₅₂₂

• X₆₉₁ = cevapoint of X₆ and X₃₅₁

• X₆₉₁ = X_i-cross conjugate of X_j for these (i,j): (23, 250), (187, 249), (351, 6)

• X₆₉₁ = circumcircle-antipode of X₈₄₂

• X₆₉₁ = anticomplement of X₅₀₉₉

• X₆₉₁ = intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,PU(2)}}

• X₆₉₁ = intersection, other than A, B, C, of circumcircle and hyperbola {A, B, C,PU(62)}}

• X₆₉₁ = Λ(X₁₁₅, X₁₂₅)

• X₆₉₁ = Ψ(X₂, X₉₉)

• X₆₉₁ = Ψ(X₆₅₀, X₂₁)

• X₆₉₁ = reflection of X₉₉ in the Euler line

• X₆₉₁ = reflection of X₁₁₀ in the Brocard axis

• X₆₉₁ = crossdifference of every pair of points on line X₁₆₄₈X₁₆₄₉

• X₆₉₁ = inverse-in-circle-O(15,16) of X₁₁₁

• X₆₉₁ = {X₁₅,X₁₆}-harmonic conjugate of X₁₁₁

• X₆₉₁ = X₁₅₇₇-isoconjugate of X₁₈₇

• X₆₉₁ = 1st-Parry-to-ABC similarity image of X₂₃

• X₆₉₁ = X₈₄₃-of-circumsymmedial-triangle

• X₆₉₁ = perspector of circumsymmedial triangle and cross-triangle of ABC and circumcevian triangle of X₁₈₇

• X₆₉₁ = perspector of circummedial triangle and cross-triangle of ABC and circumcevian triangle of X₂₃

• X₆₉₁ = barycentric product X₉₉*X₁₁₁ (circumcircle-X₂ antipodes)

Notes

Let L_A be the line of reflection of the line X₆X₁₃ in 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 X₆₉₁. (Randy Hutson, 9/23/2011)

Let A', B', C' be the intersections of line X₂X₆ and lines BC, CA, AB, resp. The circumcircles of AB'C', BC'A', CA'B' concur in X₆₉₁. (Randy Hutson, December 26, 2015)

Let A₁B₁C₁, A₂B₂C₂ and A₃B₃C₃ be the 1st, 2nd and 3rd Parry triangles. Let A' be the trilinear product A₁*A₂*A₃, and define B' and C' cyclically. The lines AA', BB', CC' concur in X₆₉₁. (Randy Hutson, February 10, 2016)

Let A'B'C' be the vertex-triangle of the 1st and 4th Brocard triangles. The lines AA', BB', CC' concur in X₆₉₁. (Randy Hutson, October 8, 2019)