Let $k$ be a field, let $E \subset \mathbb{P}^2_k$ be a nonsingular projective plane cubic, and let $O \in E(k)$ be a flex point. Define a binary operation $+$ on $E(k)$ by the chord-and-tangent construction: for $P,Q \in E(k)$, let $L_{P,Q} \subset \mathbb{P}^2_k$ be the $k$-line through $P$ and $Q$ if $P \neq Q$, and the tangent line $T_PE$ if $P=Q$; let $R$ be the residual third intersection point of $L_{P,Q}$ with $E$, counted with intersection multiplicity; then let $P+Q$ be the residual third intersection point of the $k$-line through $R$ and $O$, or of the tangent line $T_OE$ if $R=O$, again counted with intersection multiplicity. Then $P+Q \in E(k)$ for all $P,Q \in E(k)$. Moreover $O$ is an identity element, every point $P \in E(k)$ has an inverse with respect to $+$, and the operation is commutative.