X₅₈

ISOGONAL CONJUGATE OF X₁₀

X₅₈ is the isogonal conjugate of X₁₀.

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

X₅₈ (ISOGONAL CONJUGATE OF X₁₀) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {1, 21}

• Related Centers: X₁₀ (SPIEKER CENTER), X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₁ (INCENTER), X₂₁ (SCHIFFLER POINT)

Trilinears: \(a/(b + c) : b/(c + a) : c/(a + b)\)

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

Search Value (6, 9, 13): \(0.751932245362\)

▸ Alternate Coordinates7 forms

• Trilinears: \((1 - \cos A)/(\cos B + \cos C) : :\)

• Trilinears: \(sa^{2} + SR : sb^{2} + SR : sc^{2} + SR\)

• Trilinears: \(r \cos A - s \sin A : :\), where s = semiperimeter and r = inradius

• Trilinears: \(\sin(A - U) : :\), U as at X₅₇₂ and X₅₇₃

• Trilinears: \((R/r) - 1/(\cos B + \cos C) : :\)

• Trilinears: \((r/R) - \cos 2A + 1 : :\)

• Trilinears: eccentricity of A-Soddy ellipse : :

▸ Incident Central Lines37 lines

{1, 21}, {2, 540}, {3, 6}, {7, 272}, {8, 996}, {9, 975}, {10, 171}, {20, 387}, {25, 967}, {27, 270}, {28, 34}, {29, 162}, {35, 42}, {36, 60}, {40, 601}, {41, 609}, {43, 979}, {46, 998}, {56, 222}, {65, 109}, {82, 596}, {84, 990}, {86, 238}, {87, 978}, {99, 727}, {101, 172}, {103, 112}, {106, 110}, {229, 244}, {269, 1014}, {274, 870}, {314, 987}, {405, 940}, {519, 1043}, {942, 1104}, {977, 982}, {1019, 1027}

▸ Functional Relations32 relations

• X₅₈ = isogonal conjugate of X₁₀

• X₅₈ = isotomic conjugate of X₃₁₃

• X₅₈ = complement of X₁₃₃₀

• X₅₈ = anticomplement of X₃₄₅₄

• X₅₈ = circumcircle-inverse of X₁₃₂₆

• X₅₈ = Brocard-circle-inverse of X₃₈₆

• X₅₈ = X_i-Ceva conjugate of X_j for these (i,j): (81, 284), (267, 501), (270, 28)

• X₅₈ = cevapoint of X₆ and X₃₁

• X₅₈ = X_i-cross conjugate of X_j for these (i,j): (6, 81), (36, 106), (56, 28), (513, 109)

• X₅₈ = crosspoint of X_i and X_j for these (i,j): (1, 267), (21, 285), (27, 86), (60, 270)

• X₅₈ = crosssum of X_i and X_j for these (i,j): (1, 191), (6, 199), (12, 201), (37, 210), (42, 71), (65, 227), (594, 756)

• X₅₈ = crossdifference of every pair of points on line X₅₂₃X₆₆₁

• X₅₈ = X₆-Hirst inverse of X₁₃₂₆

• X₅₈ = antigonal conjugate of isogonal conjugate of X₁₃₂₄

• X₅₈ = antigonal conjugate of isotomic conjugate of X₂₁₂₇₇

• X₅₈ = antigonal conjugate of polar conjugate of X₃₇₇₇₀

• X₅₈ = X_i-beth conjugate of X_j for these (i,j): (21, 21), (60, 58), (110, 58), (162, 58), (643, 58), (1098, 283)

• X₅₈ = barycentric product of PU(31)

• X₅₈ = trilinear pole of line X₆₄₉X₈₃₄

• X₅₈ = {X₁,X₃₁}-harmonic conjugate of X₅₉₅

• X₅₈ = {X₂₁,X₂₈₃}-harmonic conjugate of X₂₃₂₈

• X₅₈ = X₄₂-isoconjugate of X₇₅

• X₅₈ = X₇₁-isoconjugate of X₉₂

• X₅₈ = X₁₀₁-isoconjugate of X₁₅₇₇

• X₅₈ = homothetic center of 2nd circumperp triangle and ‘Hatzipolakis-Brocard triangle’ (A'B'C' as defined at X₅₄₂₉)

• X₅₈ = trilinear product of vertices of 2nd circumperp triangle

• X₅₈ = perspector of 2nd circumperp triangle and unary cofactor triangle of 1st circumperp triangle

• X₅₈ = perspector of ABC and cross-triangle of ABC and 2nd circumperp triangle

• X₅₈ = Cundy-Parry Phi transform of X₅₇₃

• X₅₈ = Cundy-Parry Psi transform of X₁₃₄₇₈

• X₅₈ = perspector of ABC and unary cofactor triangle of Gemini triangle 11

• X₅₈ = {X₁,X₂₁}-harmonic conjugate of X₄₆₅₃

Notes

X₅₈ is the point of concurrence of the Brocard axes of triangles BIC, CIA, AIB, ABC, (where I denotes the incenter, X₁), as proved in Antreas P. Hatzipolakis, Floor van Lamoen, Barry Wolk, and Paul Yiu, Concurrency of Four Euler Lines, :cite:`Forum Geometricorum <forum-geometricorum>` 1 (2001) 59-68.

Let (S_a) be the reflection of the Spieker circle in BC, and define (S_b), (S_c) cyclically. X₅₈ is the radical center of (S_a), (S_b), (S_c). (Randy Hutson, July 20, 2016)

Let A'B'C' be the anticomplement of the Feuerbach triangle. Let A″B″C″ be the circumcevian triangle, wrt A'B'C', of X₁. The lines AA″, BB″, CC″ concur in X₅₈. (Randy Hutson, July 20, 2016)

Let Eu_aEu_bEu_c be the Feuerbach triangle. Let L_a be the tangent to the nine-point circle at Eu_a, and define L_b, L_c cyclically. Let A″ be the isogonal conjugate of the trilinear pole of L_a, and define B″, C″ cyclically. Let A* = BB″∩CC″, B* = CC″∩AA″, C* = AA″∩BB″. The lines AA*, BB*, CC* concur in X₅₈. (Randy Hutson, July 20, 2016)

Let A'B'C' be the 2nd circumperp triangle. Let A″ be the trilinear product B'*C', and define B″, C″ cyclically. A″, B″, C″ are collinear on line X₃₆X₂₃₈ (the trilinear polar of X₈₁). The lines AA″, BB″, CC″ concur in X₅₈. (Randy Hutson, July 20, 2016)

For an artistic design generated by X₅₈, see X₂₄₄.

X₅₈ is the {X₃,X₆}-harmonic conjugate of X₃₈₆.