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 l 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 going through P such that the resulting windmill uses each point of S as a pivot infinitely many times.