Bulletin of the Australian Mathematical Society

Research Article

Heron quadrilaterals with sides in arithmetic or geometric progression

R.H. Buchholza1 and J.A. MacDougalla2

a1 Department of Defence, Locked Bag 5076, Kingston ACT 2605 Australia e-mail: ralph@defcen.gov.au

a2 Department of Mathematics, University of Newcastle, Callaghan NSW 2308 e-mail: mmjam@cc.newcastle.edu.au

Abstract

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

(Received August 31 1998)