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Heron Quadrilaterals via Elliptic Curves



Farzali Izadi, Foad Khoshnam, Dustin Moody


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.
Rocky Mountain Journal of Mathematics


Heron quadrilaterals, cyclic quadrilaterals, congruent numbers, elliptic curves


Izadi, F. , Khoshnam, F. and Moody, D. (2017), Heron Quadrilaterals via Elliptic Curves, Rocky Mountain Journal of Mathematics, [online],, (Accessed April 13, 2024)
Created August 4, 2017, Updated October 12, 2021