Heron Quadrilaterals via Elliptic Curves

Published

August 5, 2017

Author(s)

Farzali Izadi, Foad Khoshnam, Dustin Moody

Abstract

A Heron quadrilateral is a cyclic quadrilateral with rational area. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y^2=x^3+\alpha x^2-n^2 x. This correspondence generalizes the notions of Goins and Maddox who established a similar connection between Heron triangles and elliptic curves. We further study this family of elliptic curves, looking at their torsion groups and ranks. We also explore their connection with congruent numbers, which are the \alpha=0 case. Congruent numbers are positive integers which are the area of a right triangle with rational side lengths.

Citation

Rocky Mountain Journal of Mathematics

Volume

7

Issue

4

Pub Type

Journals

Download Paper

https://doi.org/10.1216/RMJ-2017-47-4-1227

Local Download

Keywords

Heron quadrilaterals, cyclic quadrilaterals, congruent numbers, elliptic curves

Citation

Izadi, F. , Khoshnam, F. and Moody, D. (2017), Heron Quadrilaterals via Elliptic Curves, Rocky Mountain Journal of Mathematics, [online], https://doi.org/10.1216/RMJ-2017-47-4-1227, https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=917343 (Accessed October 7, 2025)