Heron Quadrilaterals via Elliptic Curves

A Heron quadrilateral is a cyclic quadrilateral whose area and side lengths are rational. In this work, we establish a correspondence between Heron quadrilaterals and a family of elliptic curves of the form y²=x³+αx²−n²x. 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 the α=0 case of congruent numbers. Congruent numbers are positive integers equal to the area of a right triangle with rational side lengths.