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.

Proof:

Consider every possible (oriented) direction of a line in the plane. For any oriented line , let denote the number of points to the right of subtracted from the number of points to the left of (since the line is oriented, left and right are well-defined). Call a direction associated with a point in if the number of points to the left of the line passing through that point with that direction satisfies It is clear that every direction is associated with a unique point in and that every point in is associated with infinitely many directions . It is also clear that as a line moves about the plane as described in the problem, the function remains invariant (except when the line passes through two points in for then it is undefined). This implies that if we choose a line with a given direction that is associated with a point in it will cycle through every point in infinitely, as desired.