Pitot's theorem, dynamic geometry and conics

Abstract

It is well known that a convex quadrilateral is a cyclic quadrilateral if, and only if, the sum of each pair of opposite angles is π. This result (which gives a necessary and sufficient condition for the existence of a circle which circumscribes a given quadrilateral) is beautifully complemented by Pitot’s theorem which says that a given convex quadrilateral has an inscribed circle if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair. Henri Pitot, a French engineer, noticed the easy part of this result in 1725 (see Figure 1), and the converse was first proved by J-B Durrande in 1815. Accordingly, we shall say that a convex quadrilateral is a Pitot quadrilateral if, and only if, the sum of the lengths of one pair of opposite edges is the same as the sum for the other pair.