Jean Poncelet was a French mathematician who substantially advanced projective geometry. With Brianchon, he proved Feuerbach's theorem on the nine-point circle in 1820–1821, and also suggested the theorem proved by Steiner and now called the Poncelot-Steiner theorem that Euclidean constructions can be done with a straightedge alone.

 

As a soldier in Napoleon's army, Poncelet was captured and imprisoned in Russia. While in prison from 1813–1814, he organized and wrote down his discoveries, and the result was published as Traité des propriétés projectives des figures (1822). To serve as an introduction to this work, he also wrote Applications d'analyse et de géométrie (2 vols., 1862–1864).

 

 

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 is Poncelet's theorem, also known as Poncelet's closure theorem.