Does every triangle have a periodic orbit?

Imagine kicking a ball across the floor of a triangular room. Every time it hits a wall it bounces off in the usual way balls bounce,  but being mathematicians let's imagine there is no curving or spin and that the ball keeps going for ever. Will it eventually come back to its orginal position, headed in its original direction? Of course, we should also imagine the ball is very small so it never gets stuck in a corner somehow. A trajectory of such a bouncing ball that eventually comes back to where it started is called a "periodic orbit". The question is: For every possible triangular-shaped room, is there some starting point and starting direction for the ball so that its trajectory is a periodic orbit? Does every triangle have a periodic orbit? Click on the image to the right to see an example.