# Poncelet, Jean Victor (1788–1867)

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**.