= Solution
Let $E$ be a smooth plane cubic whose identity $O$ is an inflection point. A line through $P$ and $Q$, using the tangent when $P=Q$, has a third intersection $R$ counted with multiplicity. The <chord-and-tangent group law> defines $P+Q$ by drawing the line through $R$ and $O$ and taking its third intersection.
The clean verification of the group axioms uses <principal divisor on an algebraic curve>[divisors]. The line at infinity meets a Weierstrass cubic in $3(O)$, so three collinear points $P,Q,R$ satisfy
$$
(P)+(Q)+(R)-3(O)=\operatorname{div}(\ell/\ell_\infty).
$$
Consequently the map
$$
E\longrightarrow\operatorname{Pic}^0(E),
\qquad P\longmapsto[(P)-(O)]
$$
sends the chord-and-tangent construction to addition of divisor classes. The <principal divisor criterion on an elliptic curve> shows that this map is bijective. Associativity and commutativity therefore follow from the <abelian group> law on $\operatorname{Pic}^0(E)$. The tangent convention handles repeated intersections, $O$ represents the zero class, and the third point on the line through $P$ and $O$ represents the inverse of $P$. Hence all group axioms hold.
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