Group law on an elliptic curve

Summary

The points of an elliptic curve $E$ over a field $K$, together with the point at infinity $O$, form an abelian group $E(K)$. The operation is the chord-and-tangent law: three points add up to $O$ exactly when they lie on a line. This group structure is what makes elliptic curves so useful, from Mordell's theorem to elliptic curve cryptography.