  In this paper, we demonstrate the intimate relationships among some geometric
figures and the families of elliptic curves with positive ranks. These
geometric figures include \textit{\textbf{Heron triangles}},
\textit{\textbf{Brahmagupta quadrilaterals}} and \textit{\textbf{Bicentric
quadrilaterals}}. Firstly, we investigate the important properties of these
figures and then utilizing these properties, we show that how to construct
various families of elliptic curves with different positive ranks having
different torsion subgroups.
