Article thumbnail

On the areas of cyclic and semicyclic polygons

By F. Miller Maley, David P. Robbins and Julie Roskies

Abstract

AbstractWe investigate the “generalized Heron polynomial” that relates the squared area of an n-gon inscribed in a circle to the squares of its side lengths. For a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a certain variety derived from the variety of binary (2m−1)-forms having m−1 double roots. Thus we obtain explicit formulas for the areas of cyclic heptagons and octagons, and illuminate some mysterious features of Robbins' formulas for the areas of cyclic pentagons and hexagons. We also introduce a companion family of polynomials that relate the squared area of an n-gon inscribed in a circle, one of whose sides is a diameter, to the squared lengths of the other sides. By similar algebraic techniques we obtain explicit formulas for these polynomials for all n⩽7

Publisher: Elsevier Inc.
Year: 2005
DOI identifier: 10.1016/j.aam.2004.09.008
OAI identifier:
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • https://s3.amazonaws.com/prod-... (external link)
  • https://s3-eu-west-1.amazonaws... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.