Re:The “Windmill” Problem on the 2011 International Mathemat...
by lelf on Thursday October 17, 2019 @02:22PM
(#21284948)
The problem:
Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line l going through a point P ∈ S. Rotate l clockwise around the pivot P until the line contains another point Q of S. The point Q now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from S.
Show that for a suitable P ∈ S and a suitable starting line l containing P, the resulting windmill will visit each point of S as a pivot infinitely often.
Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows. Start with a line l going through a point P ∈ S. Rotate l clockwise around the pivot P until the line contains another point Q of S. The point Q now takes over as the new pivot. This process continues indefinitely, with the pivot always being a point from S.
Show that for a suitable P ∈ S and a suitable starting line l containing P, the resulting windmill will visit each point of S as a pivot infinitely often.