X₃₆₀

HOFSTADTER ZERO POINT

The limit of this point as r approaches 0 is X₃₆₀. The two Hofstadter points, X₃₅₉ and X₃₆₀ are examples of transcendental triangle centers, since they have no trilinear or barycentric representation using only algebraic functions of a,b,c (or sin A, sin B, sin C).

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

X₃₆₀ (HOFSTADTER ZERO POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {2, 1115}

• Related Centers: X₂ (CENTROID), X₁₁₁₅ (EXTERIOR-ANGLE CURVATURE CENTROID)

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

Barycentrics: \(A : B : C\)

Search Value (6, 9, 13): \(1.045653663244\)

▸ Linear Combinations1 combination

• X₃₆₀ = \(A\) A + \(B\) B + \(C\) C, where A, B, C are the angles and A, B, C are the vertices

▸ Incident Central Lines1 line

{2, 1115}

▸ Functional Relations25 relations

• X₁ = Hofstadter 1/2 point = antigonal image of X₈₀

• X₃ = Hofstadter 2 point = antigonal image of X₂₆₅

• X₄ = Hofstadter -1 point

• X₃₅ = Hofstadter 3/2 point

• X₇₉ = Hofstadter -1/2 point = antigonal image of X₃₀₆₅

• X₁₈₆ = Hofstadter 3 point = antigonal image of X₅₉₆₂

• X₂₆₅ = Hofstadter -2 point = antigonal image of X₃

• X₃₅₇ = Hofstadter 1/3 point

• X₃₅₈ = Hofstadter 2/3 point

• X₃₅₉ = Hofstadter 0 point

• X₃₆₀ = Hofstadter 1 point

• X₁₁₂₇ = Hofstadter 1/4 point

• X₁₁₂₉ = Hofstadter 3/4 point

• X₅₄₅₇ = Hofstadter -1/3 point

• X₅₄₅₈ = Hofstadter -2/3 point

• X₅₉₆₁ = Hofstadter 4 point = antigonal image of X₅₉₆₄

• X₅₉₆₂ = Hofstadter -3 point = antigonal image of X₁₈₆

• X₅₉₆₃ = Hofstadter 5 point

• X₅₉₆₄ = Hofstadter -4 point = antigonal image of X₅₉₆₁

• X₃₆₀ = isogonal conjugate of X₃₅₉

• X₃₆₀ = anticomplement of X₁₁₁₅

• X₃₆₀ = X_i-anticomplementary conjugate of X_j for these (i,j): {7021, 8}, {7041, 69}

• X₃₆₀ = X_i-isoconjugate of X_j for these (i,j): {1, 359}, {6, 1077}

• X₃₆₀ = barycentric product X₇₅*X₁₀₄₉

• X₃₆₀ = barycentric quotient X_i/X_j for these {i,j}: {1, 1077}, {6, 359}, {1049, 1}, {1077, 1028}, {1085, 1049}

Notes

This point is obtained as a limit of perspectors. Let r denote a real number, but not 0 or 1. Using vertex B as a pivot, swing line BC toward vertex A through angle rB and swing line BC about C through angle rC. Let A(r) be the point in which the two swung lines meet. Obtain B(r) and C(r) cyclically. Triangle A(r)B(r)C(r) is the r-Hofstadter triangle; its perspector with ABC, called the Hofstadter r point, is the point given by trilinears

(sin rA)/sin(A - rA) : (sin rB)/sin(B - rB) : (sin rC)/sin(C - rC).

Clark Kimberling, “Hofstadter points,” Nieuw Archief voor Wiskunde 12 (1994) 109-114.

If r is an integer other than 0, 1, or 2, then the inverse-in-circumcircle of the Hofstadter r point is the Hofstadter (2-r) point;

• thus, since the isogonal conjugate of the Hofstadter r point is the Hofstadter (1-r) point, if r is not -1, 0 or 1,

• then the antigonal image of the Hofstadter r point is the Hofstadter -r point.