MathLabs

Problem 2

Let SS be a finite set of at least two points in the plane. Assume that no three points of SS are collinear. A windmill is a process that starts with a line going through a single point P∈SP\in S. The line rotates clockwise about the pivot PP until the first time that the line meets some other point belonging to SS. This point, QQ, takes over as the new pivot, and the line now rotates clockwise about QQ, until it next meets a point of SS. This process continues indefinitely. Show that we can choose a point PP in SS and a line going through PP such that the resulting windmill uses each point of SS as a pivot infinitely many times.
Step 1 of 6: Set up directions, ordinary lines and balancing lines
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In plain words

Give every line a left and right side, call a direction bad only if two points of S happen to share it, and look for a line through one point with as even a split as possible.

⌊n−12⌋\left\lfloor\frac{n-1}{2}\right\rfloor
Detailed analysis

Call the points of SS vertices and let n=∣S∣n=|S|. Direct every line so it has a well-defined right side. Call a direction ordinary if no line of that direction passes through two vertices, and call a line ordinary if its direction is ordinary; since no three vertices of SS are collinear, only finitely many directions (those of the segments joining pairs of vertices) fail to be ordinary. Call a line a balancing line if it passes through exactly one vertex and has exactly ⌊n−12⌋\left\lfloor\frac{n-1}{2}\right\rfloor of the remaining vertices to its right.