On an elliptic curve involving pairs of triangles and special quadrilaterals

Abstract

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