Reference question: Poncelet theorem A very famous theorem of Poncelet states that for an elliptic billiard all $n$-periodic trajectories are tangent to some ellipse. As far as I know, Poncelet proved this theorem while sitting in Russian jail so he didn't write it down. Could anybody give me a reference on a first book or article where the proof was actually given? 
I want to make a correct reference in the article I am writing. Moreover, I am also thinking about mentioning a greek mathematician Proclus who lived in the 5th century BC and proved that once a line in an ellipse is tangent to some other smaller ellipse, its reflection is tangent to the smaller ellipse too. 
 A: Poncelet published his theorem ("Poncelet's porism) in 1822, after he returned to France following his captivity as war prisoner in Russia:
J.V. Poncelet, Traité des propriétés projectives des figures (Paris, 1822).
The book has been scanned and can be read here or on archive.org.
A: 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.
