## X(360) (HOFSTADTER ZERO POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: tc.zip (10.1 Mb).
This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           A/a : B/b : C/c
Barycentrics    A : B : C

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 is the point given by trilinears

sin(r(A))/sin(A - r(A)) : sin(r(B))/sin(B - r(B)) : sin(r(C))/sin(C - r(C)).

The limit of this point as r approaches 0 is X(360). The two Hofstadter points, X(359) and X(360) 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).

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

X(360) = isogonal conjugate of X(359)
X(360) = anticomplement of X(1115)

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense