2011 IMO Shortlist Problems/C3

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