Generalization of Heron\u2019s and Brahmagupta\u2019s equalities to any cyclic polygon

Paolo Dulio,Enrico Laeng

doi:10.1007/s00010-020-00771-w

Abstract

It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. 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. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n>4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

Highlights

A natural and largely considered question in convex geometry is the determination of the area A of a convex polygon as a symmetric function of the lengths of its edges
Several results concerning the geometry of cyclic polygons have been obtained in different areas of research, which points to a general interest in such geometric objects
We have shown that Heron’s and Brahmagupta’s equalities can be extended to a formula that provides the square of the area of any convex cyclic polygon as a symmetric polynomial of the lengths of the segments determined on the edges by the incircles of a suitable triangulation

Summary

Introduction

A natural and largely considered question in convex geometry is the determination of the area A of a convex polygon as a symmetric function of the lengths of its edges. This has proved to be far from simple. It was observed that Heron and Brahmagupta’s formulae can be restated in a form where 16A2 represents a monic polynomial whose coefficients are symmetric polynomials in the squares of the edges This generalizes to cyclic pentagons and hexagons, where polynomials of degree 7 and 38 respectively appear, but the formulae, even if they hold in the non convex case as well, do not provide explicit forms for the area (see [7] for interesting comments and remarks).

Notations and preliminaries

Heron’s equality

Brahmagupta’s equality

The area of a circular polygon having an arbitrary number of edges

Conclusion and remarks