Poncelet's theorem

Given an ellipse, and a smaller ellipse entirely inside it, start at a point on the outer ellipse, and, moving clockwise (say), follow a line that is tangent to the inner ellipse until you hit the outer ellipse again. Repeat this over and over again. It may be that this path will never hit the same points on the outer ellipse twice. However, if it does close up in a certain number of steps, then something amazing is true: all such paths, starting at any point on the outer ellipse, close up in the same number of steps. This fact is Poncelet's theorem, also known as Poncelet's closure theorem, and is named after the Jean Poncelet.