Poncelet Porism in Ellipses
Poncelet's porism is an assertion that provided there is a polygon circumscribed around one conic and inscribed into another, there are infinitely many such polygons. For two circles, one nested in the other, a demonstration is available elsewhere.
The applet below illustrates Poncelet's porism in a more general situation of two ellipses, one inside the other.
Denote the bigger ellipse Γ and the smaller one γ. Choose point A on Γ and find a tangent from A to γ. Extend this tangent through the intersection with Γ at, say, B. For B, repeated the process of finding the tangent to γ and extending it to the intersection with Γ. Let the process go on. It is quite possible that one of the tangents will pass through the starting point A, thus closing a polygon inscribed in Γ and circumscribed around γ. Such a polygon is sometimes referred to as a Poncelet's traverse. The essence of Poncelet's porism is an assertion that if a traverse exists for one point A, a traverse will exist for any starting point A.
The porism is illustrated in the cases of a triangle and a quadrilateral. There are two ellipse: a blue within a red one. Both are defined by their foci and an extra point. On the red ellipse there is a solid point that is to serve as a vertex of a polygon - the meeting place of the first and the last tangents from one ellipse to the other. First adjust the ellipses as to close the polygon of the tangents and then drag the solid point.