Existence of periodic trajectories in obtuse triangle billiards

Determine whether planar billiards within a general obtuse triangle admit any periodic trajectories.

Background

The paper reviews periodic trajectories in triangular billiards: in acute triangles, the inscribed triangle formed by the feet of the altitudes is the unique 3-periodic billiard trajectory of minimal perimeter, and right triangles admit explicit periodic examples. These cases are established using classical geometric arguments and the billiard reflection law.

In contrast, for general obtuse triangles, the existence of periodic billiard trajectories is not settled. The authors note special examples and reference computer-based proofs establishing periodic trajectories when the obtuse angle does not exceed 100 degrees, leaving the general case unresolved.

References

After such elementary considerations for acute and right triangles, one can stay amazed by the fact that it is not known if billiards within general obtuse triangles have any periodic trajectories! There are examples for some special cases and also an intriguing computer-based proof for the existence of periodic billiard trajectories when the obtuse angle does not exceed 100 , see [Sch2006,Sch2009].

— Is every triangle a trajectory of an elliptical billiard?  (2405.08922 - Dragović et al., 2024) in Section 2.1 (Triangular billiards), paragraph following Figure 7