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 L going through a single point P ∈ 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.

Question

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 L going through a single point P ∈ 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.

turingtest · Answer

Show that we can choose a point P in S and a line L going through P such that the resulting windmill uses each point of S as a pivot inﬁnitely many times.

asnaseer · Answer

Lets say that we start with this situation with 3 points in S named P, Q and R and we hapen to start by selecting $l$ such that it does not pass through the region enclosed by PQR:
|dw:1323302062225:dw|

asnaseer · Answer

after one clockwise sweep we would get:

asnaseer · Answer

|dw:1323302282313:dw|
which sweeps the plane S until the line $l$ hits Q.