X₈₅₀

BARYCENTRIC MULTIPLIER FOR KIEPERT HYPERBOLA

The barycentric product of X₈₅₀ and the circumcircle is the Kiepert hyperbola.

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

X₈₅₀ (BARYCENTRIC MULTIPLIER FOR KIEPERT HYPERBOLA) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {99, 476}, {2, 647}

• Related Centers: X₁₁₀ (FOCUS OF KIEPERT PARABOLA), X₉₉ (STEINER POINT), X₄₇₆ (TIXIER POINT), X₂ (CENTROID), X₆₄₇ (CROSSDIFFERENCE OF X₂ AND X₃)

Trilinears: \((b^{2} - c^{2})/a^{3} : :\)

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

Search Value (6, 9, 13): \(18.041819248310\)

▸ Alternate Coordinates2 forms

• Barycentrics: \(\csc A \sin(B - C) : :\)

• Barycentrics: \(|AP(1)|^{2} - |AU(1)|^{2} : :\)

▸ Incident Central Lines9 lines

{2, 647}, {99, 476}, {110, 685}, {297, 525}, {316, 512}, {325, 523}, {340, 520}, {669, 804}, {670, 892}

▸ Functional Relations15 relations

• X₈₅₀ = isotomic conjugate of X₁₁₀

• X₈₅₀ = complement of X₃₁₂₉₆

• X₈₅₀ = anticomplement of X₆₄₇

• X₈₅₀ = anticomplementary conjugate of X₃₉₃₅₂

• X₈₅₀ = X_i-Ceva conjugate of X_j for these (i,j): (2, 36901), (76, 338), (99, 311), (264, 339)

• X₈₅₀ = X_i-cross conjugate of X_j for these (i,j): (115, 1502), (125, 2), (338, 76), (339, 264)

• X₈₅₀ = crosspoint of X₉₅ and X₉₉

• X₈₅₀ = crosssum of X_i and X_j for these (i,j): (32, 669), (39, 647), (51, 512)

• X₈₅₀ = crossdifference of every pair of points on line X₃₂X₁₈₄

• X₈₅₀ = isogonal conjugate of X₁₅₇₆

• X₈₅₀ = trilinear pole of line X₁₁₅X₁₂₇ (polar of X₁₁₂ wrt polar circle)

• X₈₅₀ = pole wrt polar circle of trilinear polar of X₁₁₂ (line X₆X₂₅)

• X₈₅₀ = polar conjugate of X₁₁₂

• X₈₅₀ = intersection of trilinear polars of X₃₀₀ and X₃₀₁

• X₈₅₀ = barycentric product X₇₆*X₅₂₃

Notes

Let P₁ and P₂ be the two points on the de Longchamps line whose trilinear polars are parallel to the de Longchamps line. P₁ and P₂ lie on the Kiepert hyperbola and circle {X₂, X₉₈, X₉₉}}. X₈₅₀ is the barycentric product P₁*P₂. (Randy Hutson, June 7, 2019)