ISL 1970 Problem 40

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.