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Poncelet's closure theorem
In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem) states that whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.〔Weisstein, Eric W. "Poncelet's Porism." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PonceletsPorism.html〕 It is named after French engineer and mathematician Jean-Victor Poncelet.
Poncelet's porism can be proved by an argument using an elliptic curve, whose points represent a combination of a line tangent to one conic and a crossing point of that line with the other conic.
==Statement==
Let ''C'' and ''D'' be two plane conics. If it is possible to find, for a given ''n'' > 2, one ''n''-sided polygon that is simultaneously inscribed in ''C'' (meaning that all of its vertices lie on ''C'') and circumscribed around ''D'' (meaning that all of its edges are tangent to ''D''), then it is possible to find infinitely many of them. Each point of ''C'' or ''D'' is a vertex or tangency (respectively) of one such polygon.
if the conics are circles, the polygons that are inscribed in one circle and circumscribed about the other are called bicentric polygons, so this special case of Poncelet's porism can be expressed more concisely by saying that every bicentric polygon is part of an infinite family of bicentric polygons with respect to the same two circles.〔Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publications, 2007 (orig. 1960).〕
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