While I was reading Tanya Khovanova‘s blog I came across this problem which Andrei Zelevinsky reportedly put in his list of important problems that undegraduate students should think of. The problem reads:
Consider a procedure: Given a polygon in a plane, the next polygon is formed by the centers of its edges. Prove that if we start with a polygon and perform the procedure infinitely many times, the resulting polygon will converge to a point.
I’d like to present one solution in the rest of this post.
One way to solve this problem is to use complex coordinates. Denote by the points. One iteration of the process yields the points . More generally, the -th iteration yields the points , where
Now we can prove that the tend to as , independently of and . To see this let us take the example of , WLOG. A straightforward calculation using the -th roots of unity gives ;
This means that the procedure converges to a unique point which is the barycenter of the original points, which concludes the proof.
In the next variation, instead of using the centers of edges to construct the next polygon, use the centers of gravity of consecutive vertices.