Problem 2

Let

S

be a finite set of at least two points in the plane. Assume that no three points of

S

are collinear. A windmill is a process that starts with a line going through a single point

P\in S

. The line rotates clockwise about the pivot

P

until the first time that the line meets some other point belonging to

S

. This point,

Q

, takes over as the new pivot, and the line now rotates clockwise about

Q

, until it next meets a point of

S

. This process continues indefinitely. Show that we can choose a point

P

S

and a line going through

P

such that the resulting windmill uses each point of

S

as a pivot infinitely many times.

Step 2 of 6: Build a balancing line through each vertex

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In plain words

Spin a line 180° about a fixed vertex: the count of points on its right changes by exactly one point at a time, so it must pass through the balanced value on the way.

0\to k\to n-1-k

Detailed analysis

Fix a vertex $P$ and start with any ordinary line through $P$ , say with $k$ points to its right. As the line rotates $180^\circ$ about the pivot $P$ , each of the other $n-1$ vertices $Q$ switches sides exactly once, precisely when the line's direction passes through the (distinct, since no three points are collinear) direction of $PQ$ ; after the full half-turn the right-count has gone from $k$ to $n-1-k$ , changing by exactly $1$ at each such crossing. Hence at some ordinary direction along the way the right-count equals $\left\lfloor\frac{n-1}{2}\right\rfloor$ , and the line at that moment is the desired ordinary balancing line through $P$ .