Just an animation. Let it run
Let {S} be a finite set of at least two points in the plane. Assume that no three points of S are collinear. A windmill is a process that starts with a line E going through a single point P in S. The line rotates clockwise about the pivot P until the first time that the line meets some other point belonging to 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 S. This process continues indefinitely. Show that we can choose a point S in 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.
