Poncelet's Closure Theorem
In geometry Poncelet's porism 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. It is named after the French engineer and mathematicion Jean-Victor Poncelet.
His 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.