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Set of all directions in which the ball will move is finite

Source: IMO LongList 1970 - P40

May 22, 2011
geometrygeometric transformationreflectioncombinatorics proposedcombinatorics

Problem Statement

Let ABC be a triangle with angles α,β,γ\alpha, \beta, \gamma commensurable with π\pi. Starting from a point PP interior to the triangle, a ball reflects on the sides of ABCABC, 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,CA,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 00 to infinity consists of segments parallel to a finite set of lines.
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