An Unexpected Maximum in a Family of Rectangles

The reader is encouraged to try to solve the problem now, before proceeding. In our solution to this problem we realized that A1 A2 A3A4 need not be a square. This observation led to the study of families of rectangles that satisfy hypotheses like those of Problem 749, and ultimately to this paper. A quick solution to Problem 749 is provided by Theorem 1, which is a special case of Feuerbach's Relation (see [4] or [6]).