abstract
On periodic billiard trajectories in obtuse triangles
Lorenz Halbeisen and Norbert Hungerbühler
In 1775, J.F. de Tuschis a Fagnano observed that in every acute
triangle, the orthoptic triangle represents a periodic billiard
trajectory, but to the present day it is not known whether or not in
every obtuse triangle a periodic billiard trajectory exists.
The limiting case of right triangles was settled in 1993 by F. Holt
who proved that all right triangles possess periodic trajectories.
The same result appeared already in 1991 independently in the Russian
literature, namely in the work of G.A. Gal'perin, A.M. Stepin
and Y.B. Vorobets. The latter authors discovered in 1992 a class
of obtuse triangles which contain particular periodic billiard paths.
In this article, we review the mentioned results and some of the
techniques used in the proofs, and at the same time show for an
extended class of obtuse triangles that they contain periodic
billiard trajectories.