IMO 2011 Problem 2

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$ . The line rotates clockwise about the pivot $P$ until the first time that the line meets some other point belonging to $\mathcal S$ . This point, $Q$ , takes over as the new pivot, and the line now rotates clockwise about $Q$ , until it next meets a point of $\mathcal S$ . This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times.

Proposed by Geoffrey Smith, United Kingdom