Associativity of the chord-and-tangent law via divisor classes

For a smooth genus one curve with origin $O$, the map $P\mapsto [(P)-(O)]$ identifies geometric points with the degree-zero Picard group of a curve. Riemann-Roch theorem gives a unique effective representative for each divisor class of degree one: its Riemann-Roch space has dimension one. A line cuts a plane cubic in a divisor linearly equivalent to $3(O)$ when $O$ is the Weierstrass flex. The two successive line intersections in the chord-and-tangent group law therefore give addition of the corresponding divisor classes. Associativity follows by injectivity of the identification and associativity of divisor-class addition, including tangent intersection multiplicities.