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 6 of 6: Conclude: every vertex is a pivot infinitely often

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In plain words

The special direction built through P can only ever be realized by the balancing line through P, and that direction keeps coming back forever.

P

Detailed analysis

Let $\delta_P$ be the direction of the ordinary balancing line through $P$ built earlier, and start the windmill from that line. By the invariant, every ordinary line occurring in this windmill is a balancing line, and by the previous step the only balancing line with direction $\delta_P$ is the one through $P$ ; so whenever direction $\delta_P$ recurs in the windmill, $P$ must be the pivot at that moment. Since $\delta_P$ recurs after every subsequent $180^\circ$ of rotation, $P$ is the pivot infinitely often. As $P$ was an arbitrary vertex of $S$ , we may in particular start the windmill from the balancing line through any chosen point, and the same argument shows every vertex of $S$ is used as a pivot infinitely many times.