X₂₅₀₁

RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, TAYLOR CIRCLE}

X₂₅₀₁ = perspector of circumconic centered at X₁₃₆

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

X₂₅₀₁ (RADICAL CENTER OF {CIRCUMCIRCLE, NINE-POINT CIRCLE, TAYLOR CIRCLE}) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {230, 231}, {112, 476}

• Related Centers: X₂₃₀ (-CEVA CONJUGATE OF X₁₁₄), X₂₃₁ (-CEVA CONJUGATE OF X₁₂₈), X₁₁₂ (Ψ(ORTHOCENTER, SYMMEDIAN POINT)), X₄₇₆ (TIXIER POINT)

Trilinears: \(\tan A \sin(B - C) : :\)

Barycentrics: \(S_{B}\)\(S_{C}(S_{B} - S_{C})\) : \(S_{C}\)\(S_{A}(S_{C} - S_{A})\) : \(S_{A}\)\(S_{B}(S_{A} - S_{B})\)

Search Value (6, 9, 13): \(-2.124582703505\)

▸ Alternate Coordinates1 form

• Trilinears: \((b^{2} - c^{2})\sec A : (c^{2} - a^{2})\sec B : (a^{2} - b^{2})\sec C\)

X₂₅₀₁ lies on the inscribed parabola having focus X₁₁₂.

▸ Incident Central Lines12 lines

{4, 1499}, {25, 669}, {107, 685}, {112, 476}, {230, 231}, {297, 525}, {393, 2433}, {419, 3288}, {421, 2623}, {450, 2451}, {460, 512}, {648, 892}

▸ Functional Relations24 relations

• X₂₅₀₁ = isogonal conjugate of X₄₅₅₈

• X₂₅₀₁ = isotomic conjugate of X₄₅₆₃

• X₂₅₀₁ = cevapoint of X_i and X_j for these (i,j): (6, 2079), (512, 2489)

• X₂₅₀₁ = crosspoint of X_i and X_j for these (i,j): (2, 925), (4, 648), (107, 2052)

• X₂₅₀₁ = crosssum of X_i and X_j for these (i,j): (3, 647), (6, 924), (343, 525), (520, 577), (906, 1331)

• X₂₅₀₁ = X_i-Ceva conjugate of X_j for these (i,j): (2, 136), (92, 2969), (107, 25), (112, 53), (115, 393), (264, 2971), (393, 115), (512, 523), (648, 4), (653, 407), (685, 460), (687, 30), (1783, 1865), (1897, 430), (2052, 2970), (2971, 264)

• X₂₅₀₁ = perspector of circumconic centered at X₁₃₆

• X₂₅₀₁ = center of circumconic that is locus of trilinear poles of lines passing through X₁₃₆

• X₂₅₀₁ = X₂-Ceva conjugate of X₁₃₆

• X₂₅₀₁ = radical center of {circumcircle, nine-point circle, orthosymmedial circle}

• X₂₅₀₁ = radical center of {circumcircle, orthocentroidal circle, orthosymmedial circle}

• X₂₅₀₁ = radical center of {nine-point circle, orthocentroidal circle, orthosymmedial circle}

• X₂₅₀₁ = intersection of orthic axes of ABC and orthic triangle

• X₂₅₀₁ = trilinear pole of line X₁₁₅X₂₉₇₁ (polar of X₉₉ wrt polar circle)

• X₂₅₀₁ = pole wrt polar circle of trilinear polar of X₉₉ (line X₂X₆)

• X₂₅₀₁ = polar conjugate of X₉₉

• X₂₅₀₁ = perspector, wrt anticevian triangle of X₄, of bianticevian conic of X₁ and X₄

• X₂₅₀₁ = crossdifference of every pair of points on line X₃X₄₉

• X₂₅₀₁ = X₆₅₀-of-orthic-triangle if ABC is acute

• X₂₅₀₁ = excentral-to-ABC functional image of X₆₅₀

• X₂₅₀₁ = PU(4)-harmonic conjugate of X₂₃₀

• X₂₅₀₁ = excentral-to-ABC functional image of X₆₅₀

• X₂₅₀₁ = barycentric product X₄*X₅₂₃

• X₂₅₀₁ = barycentric product X₂₄₀₀₇*X₂₄₀₀₈ (the Kiepert hyperbola intercepts of the orthic axis)

Notes

Let L_A be the orthic axis of triangle BCX₄, and define L_B, L_C cyclically. Let A' = L_B∩L_C, B' = L_C∩L_A, C' = L_A∩L_B. Triangle A'B'C' is here named the orthic axes triangle. A'B'C' is also the cevian triangle of X₄ wrt the orthic triangle. A'B'C' is perspective to ABC at X₄, and homothetic to the polar triangle of the nine-point circle at X₄₂₇. The orthic axes of ABC, the orthic triangle, and the orthic axes triangle concur in X₂₅₀₁. (Randy Hutson, August 19, 2019)

Let A″B″C″ be the reflection triangle. Then A' is the polar-circle-inverse of A″. (Randy Hutson, January 19, 2020)