An axiomatic look at a windmill (Q2510078)

At the 52nd International Mathematical Olympiad, the following problem was posed. Let \(\mathcal S\) be a finite set of at least two points in the plane no three of which are collinear. A windmill is a process which starts with a line \(l\) through some point \(P \in \mathcal S\) and rotates it clockwise around \(P\) until it meets the first other point \(Q \in \mathcal S\). Then the line is rotated around \(Q\), and this process is iterated indefinitely. One shall show that it is possible to choose \(P\) and \(l\) in such a way that each point of \(\mathcal S\) arises infinitely often as rotation center. The author gives an axiomatic description of this process in the framework of ordered incidence geometry and shows that the result is valid in ordered regular incidence planes, the weakest ordered geometries which can be embedded in ordered projective planes.