Hi,

Using the isomorphism between an elliptic curve $E$ and its $Pic_1(E)$ group, one can easily give $E$ the structure of a group variety after choosing a point $O\in E$. The operation that one gets is: $$P+Q+R = 0\text{ in $E$ iff }P+Q+R-3O\text{ (as divisors) is a principal divisor}.$$

- Question: Why is the condition $P+Q+R-3O$ being a principal divisor equivalent to $P+Q+R$ being the intersection divisor of $E$ with a line?

Thanks!