A "right" path to cyclic polygons

Abstract

It is well known that Heron's theorem provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its sides. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges, which, surprisingly, has revealed to be far from simple. In this paper we investigate such a problem by following a new and elementary approach. We start from the simple observation that the incircle of a right triangle touches its hypothenuse in a point that splits it into two segments, the product of whose lengths equals the area of the triangle. From this curious fact we derive in a few lines: an unusual proof of the Pythagoras' theorem, Heron's theorem for right triangles, Heron's theorem for general triangles, and Brahmagupta's theorem for cyclic quadrangles. This suggests that cutting the edges of a cyclic polygon by means of suitable points should be the "right" working method. Indeed, following this idea, we obtain an explicit formula for the area of any convex cyclic polygon, as a symmetric function of the segments split on its edges by the incircles of a triangulation. We also show that such a symmetry can be rediscovered in Heron's and Brahmagupta's results, which consequently represent special cases of the general provided formula.