X₃₅₉

HOFSTADTER ONE POINT

The Hofstadter ellipse E(1/2), given by \(x^{2}\) + \(y^{2}\) + \(z^{2}\) - 2yz - 2xz - 2xy = 0, has center X₃₇ and passes through X_i for these i: 244, 678, 2310, 2632, 2638, 2643, 3248, 4094, 4117, 10501, 23063, 24012.

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

X₃₅₉ (HOFSTADTER ONE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Related Centers: X₂ (CENTROID)

Trilinears: \(a/A : b/B : c/C\)

Barycentrics: \(a^{2}/A : b^{2}/B : c^{2}/C\)

Search Value (6, 9, 13): \(2.333647742736\)

▸ Incident Central Lines1 line

{6, 9036}

▸ Functional Relations6 relations

• X₃₅₉ = isogonal conjugate of X₃₆₀

• X₃₅₉ = X₂-Ceva conjugate of X₅₉₄₅

• X₃₅₉ = X_i-isoconjugate of X_j for these (i,j): {1, 360}, {2, 1049}

• X₃₅₉ = barycentric product X_i*X_j for these {i,j}: {1, 1077}, {1028, 1049}

• X₃₅₉ = barycentric quotient X_i/X_j for these {i,j}: {6, 360}, {31, 1049}, {1077, 75}

• X₃₅₉ = perspector of Hofstadter 0-ellipse

Notes

This point is the limit as r approaches 1 of the Hofstadter r point. See X₃₆₀ for details.

\(x^{2}\) + \(y^{2}\) + \(z^{2}\) + yz(D + 1/D) + zx(E + 1/E) + xy(F + 1/F) = 0, where D = cos A - sin A cot rA, E = cos B - sin B cot rB, F = cos C - sin C cot rC.

Taking the limit of E(r) as r tends to 0 gives information about the circumellipse, E(0) (which is also E(1)):

Equation: ayz/A + bzx/B + cxy/C = 0

Center: X₅₉₄₅ = (a/A)(\(b^{2}\)/B + \(c^{2}\)/C - \(a^{2}\)/A) : (b/B)(\(c^{2}\)/C + \(a^{2}\)/A - \(b^{2}\)/B) : (c/C)(\(a^{2}\)/A + \(b^{2}\)/B - \(c^{2}\)/C)

Intersection with circumcircle (other than A, B, C): X₃₀₆₇ = a/[A(B - C)] : b/[B(C - A)] : c/[C(A - B)]

The Hofstadter ellipse E(1/2) is also the incentral inellipse, centered at X₃₇, and touching ABC at the traces of X₁. The ellipse also passes through X₄₀₉₄, and is the incentral isotomic conjugate of line X₅₁₂X₄₈₉₅. (Randy Hutson, October 15, 2018)

See Hofstadter Ellipse at MathWorld.

See also the preamble just before X₄₂₀₇₄.