Hofstadter points


In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point and Hofstadter one-point, are particularly interesting. They are two transcendental triangle centers. Hofstadter zero-point is the center designated as X and the Hofstafter one-point is the center denoted as X in Clark Kimberling's Encyclopedia of Triangle Centers. The Hofstadter zero-point was discovered by Douglas Hofstadter in 1992.

Hofstadter triangles

Let ABC be a given triangle. Let r be a positive real constant.
Rotate the line segment BC about B through an angle rB towards A and let LBC be the line containing this line segment. Next rotate the line segment BC about C through an angle rC towards A. Let L'BC be the line containing this line segment. Let the lines LBC and L'BC intersect at A. In a similar way the points B and C are constructed. The triangle whose vertices are A, B, C is the Hofstadter r-triangle of triangle ABC.

Special case

The trilinear coordinates of the vertices of the Hofstadter r-triangle are given below:

Hofstadter points

For a positive real constant r > 0, let A B C be the Hofstadter r-triangle of triangle ABC. Then the lines AA, BB, CC are concurrent. The point of concurrence is the Hofstdter r-point of triangle ABC.

Trilinear coordinates of Hofstadter ''r''-point

The trilinear coordinates of Hofstadter r-point are given below.

Hofstadter zero- and one-points

The trilinear coordinates of these points cannot be obtained by plugging in the values 0 and 1 for r in the expressions for the trilinear coordinates for the Hofstdter r-point.
Trilinear coordinates of Hofstadter zero-point
Trilinear coordinates of Hofstadter one-point