# 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.