Polygons Formed by Perpendicular Bisectors

A curious problem that was posted in the American Mathematical Monthly in 1953 waited for a solution for more than 40 years. Meanwhile it was written and talked about as an example of an unproven problem verified by the dynamic geometry software. Along the way several generalizations had been suggested.

Given a polygon Q = Q₀, the perpendicular bisectors of its sides form another polygon Q₁. In turn, the perpendicular bisectors of Q₁ form a polygon Q₂, and so on. It has been shown in 1997 that for quadrilaterals, Q₂ is homothetic to Q. The question is, does their exist any relation between the n-gons Q_i, i = 0, 1, ..., where n > 4?

The applet below might help in investigating that question. Good luck.

References

1. B. Grünbaum, Quadrangles, Pentagons, and Computers, Geombinatorics 3 (1993), pp. 4-9

Copyright © 1996-2009 Alexander Bogomolny