Billiard Paths Must End:If the dimensions of a rectangle are positive integers, the path from any corner must end at one of the other corners. Proof: The path must always hit sides an integral distance from the ends of the sides. A path cannot hit a side twice in the same location. Since there are only a finite number of places where a path can hit on each side, the path must be finite. And of course, the only way a path can end is at a corner. Note that the path can't end at the corner at which it started---there is no way for a path to turn around and retrace its steps. |