Poncelet's Porism
Try this experiment. Each of the two circles can be moved and can be modified by dragging its center or the blue point on its circumference. There's a red point on the larger circle from which a tangent is drawn to the smaller circle. The tangent intersects the big circle at some point from where another tangent to the small circle is drawn, and so on. There may be quite a few such tangents. (To see many, make the radii of the two circles nearly equal.)
Play with the circles and the blue points. This may happen that the end of the last tangent will fall exactly in the red point, which will create a closed broken line - a polygon. The eyes open when this happens. Now drag the red point. The polygon will change, but will nonetheless remain inscribed into the big circle and circumscribed around the small one. The line will not break.
This fact is known as Poncelet's porism: If two circles are such that there's a polygon inscribed in one and circumscribed around the other, there are infinitely many such polygons. The word "porism" signifies exactly this fact. Either there are no solutions (polygons simultaneously inscribed and circumscribed) or there are infinitely many such solutions. For this reason, Steiner's chain is also known as Steiner's porism.
As a matter of fact, the same phenomenon holds for any two conics. For two nested ellipses a demonstration is available elsewhere as is a proof for a triangle (a three sided polygon.)
