IMO 1970 LL POL40

Origin: POL

Problem

Let ABC be a triangle with angles \alpha, \beta, \gamma commensurable with \pi. Starting from a point P interior to the triangle, a ball reflects on the sides of ABC, respecting the law of reflection that the angle of incidence is equal to the angle of reflection. Prove that, supposing that the ball never reaches any of the vertices A, B, C, the set of all directions in which the ball will move through time is finite. In other words, its path from the moment 0 to infinity consists of segments parallel to a finite set of lines.