  Given any positive integer n, it is well known that there always exist
triangles with rational sides a, b and c such that the area of the triangle is
n. Assuming finiteness of the Shafarevich-Tate group, we first construct a
family of infinitely many Heronian elliptic curves of rank exactly 1 from Heron
triangles of a certain type. We also explicitly produce a separate family of
infinitely many Heronian elliptic curves with 2-Selmer rank lying between 1 and
3.
