Poncelet's closure theorem

[1][2] It is named after French engineer and mathematician Jean-Victor Poncelet, who wrote about it in 1822;[3] however, the triangular case was discovered significantly earlier, in 1746 by William Chapple.

[4] 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.

For simplicity, assume that C and D meet transversely (meaning that each intersection point of the two is a simple crossing).

Then by Bézout's theorem, the intersection C ∩ D of the two curves consists of four complex points.

Any involution of an elliptic curve with a fixed point, when expressed in the group law, has the form x → p − x for some p, so

Similarly, the projection X → D is a degree 2 morphism ramified over the contact points on D of the four lines tangent to both C and D, and the corresponding involution