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 11, 2025)
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Heron Quadrilaterals via Elliptic Curves
Published
Author(s)
Farzali Izadi, Foad Khoshnam,
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
Keywords
Heron quadrilaterals, cyclic quadrilaterals, congruent numbers, elliptic curves
Citation
Issues
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Created August 4, 2017, Updated October 12, 2021