On an elliptic curve involving pairs of triangles and special quadrilaterals

In this paper, we give an elliptic curve to obtain that there exist infinitely many rational theta-triangle and omega-trapezoid (omega-cyclic quadrilateral or perpendicular quadrilateral) pairs with areas and perimeters in fixed proportions (alpha, beta), respectively, where alpha and beta are positive rational numbers. Moreover, we get that the sets of the triples of rational theta-triangle, omega-trapezoid and omega-cyclic quadrilateral and the quadruples of rational theta-triangle, right trapezoid, pi-2-cyclic quadrilateral and perpendicular quadrilateral, with the same area and the same perimeter respectively, are infinite.