Let $C$ be a nonsingular cubic curve in $\mathbb{P}^2(\mathbb{C})$ and $P$ a point in $C$. Consider the well known group structure on $C$ with its binary operation $+$. Assume that the neutral element is chosen to be an inflection point.
How to see (preferably as elementary as possible) that there is a point $Q$ in $C$ so that $Q+Q=P$? What about the existence of a point $R_n$ in $C$ so that $nR_n = P$ with $n\geq 3$?
Given any point in the plane, there is at least one line through that point which is tangential to your curve. Now, take the point $-P\in C$ and draw a line through $-P$ tangential to $C$, say at $Q$. Then $-P+2Q=0$, solving your problem for $2Q=P$.