There seems to be a misconception here: Poncelet Theorem (at least the great one, which I believe to remember is the one he proved while in jail) is a much deeper, and more difficult statement than what you state.

Consider an ellipse inside another ellipse, and play inner-outer billiard with them. This means that you start from a point on the outer ellipse, choose one of the two line from this point tangent to the inner ellipse, and take the second intersection point of this line with the outer ellipse. You continue, always taking the next line tangent to the inner ellipse from the current point, and the other intersection point with the outer ellipse from the current line.

Theorem (Poncelet) $-$ If one orbit of this dynamical system is periodic, then all orbits are periodic.

This, if I remember well, is in Berger's Geometry. There might be a reference there.