Windmill
geometry
Source: IMO 2011, C3
The Problem
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. By a windmill we mean a process as follows. Start with a line \ell going through a point P \in \mathcal{S}. Rotate \ell clockwise around the pivot P until the line contains another point Q of \mathcal{S}. The point Q now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from \mathcal{S}.
Show that for a suitable P \in \mathcal{S} and a suitable starting line \ell containing P, the resulting windmill will visit each point of \mathcal{S} as a pivot infinitely often.
Spoiler: a neat 3blue1brown video and lesson on the solution.
