Given a billiard trajectory on a regular polygon starting from the
center of the polygon and eventually ending at a vertex, is it true that
the trajectory starting from the center but in the opposite direction also ends eventually at a vertex ?

By symmetry, this is certainly true if the number of sides is even. In the odd case, we will see that the answer is 'yes' for the equilateral triangle and the regular pentagon, but 'no' if the polygon has 7 sides or more!

The question is closely related to determining the real numbers (in some algebraic field) having a finite Hecke continued fraction expansion, or equivalently cusp representative of the Hecke modular surface. We will also encounter so-called translation surfaces and their Veech groups, and we will discuss the notion of connection points on such surfaces.