On Periodic Billiard Trajectories in Obtuse Triangles

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 had appeared independently in the Russian literature in 1991, 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 above-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.