The International Mathematical Olympiad (IMO)  2011
Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows.

Start with a line ℓ going through a point P ∈ S

Rotate ℓ clockwise around the pivot P until the line contains another point Q of S.

The point Q now takes over as the new pivot.
This process continues indefinitely, with the pivot always being a point from S.
Show that for a suitable P ∈ S and a suitable starting line ℓ containing P, the resulting windmill will visit each point of S as a pivot infinitely often.