Integral isosceles triangle–parallelogram and Heron triangle–rhombus pairs with a common area and common perimeter

Abstract

In this paper we show that there are infinitely many pairs of integer isosceles triangles and integer parallelograms with a common (integral) area and common perimeter. We also show that there are infinitely many Heron triangles and integer rhombuses with common area and common perimeter. As a corollary, we show there does not exist any Heron triangle and integer square which have a common area and common perimeter.

A family of elliptic curves of rank ≥ 5 over Q(m)

“…By the specialization theorem of Silverman [18], in order to prove that the family of elliptic curves E(t) has rank at least three over Q(t), it suffices to find a specialization t = t 0 such that the points P i (t), i = 1, 2, 3, are linearly independent on a specialized curve over Q. If we take t = 2, then the points: P 1 (2) = (0, 4) , P 2 (2) = (−3, 8) , P 3 (2) = (5,4) are linearly independent points of infinite order on the elliptic curve E(2) : y 2 = x 3 − 25x + 16. Indeed, the regulator, i.e., the determinant of the Néron-Tate height pairing matrix of these points is the non-zero value 2.94853892225094, according to SAGE [17].…”

“…is of rank ≥ 3 for infinitely many rational t. Notably, the curves (1) over Q(m) also appear in some of recent works dealing with geometric problems, see, for example, [5,21,22]. In 2017, Das et al [5] showed that the rank of the elliptic curve…”

Heron triangle and rhombus pairs with a common area and a common perimeter

“…As before, we only need to consider the rational isosceles triangle and θ-rational rhombus pairs. As in [3], we may take the equal legs of the isosceles triangle to have length u 2 + v 2 , with the base being 2(u 2 − v 2 ) and the altitude 2uv, for some rational u, v. The area of the isosceles triangle is 2uv(u 2 −v 2 ), with an perimeter of 4u 2 .…”

“…At the same year, S. Chern [2] proved that there are infinitely many integral right triangle and θ-integral rhombus pairs. In a recent paper, P. Das, A. Juyal and D. Moody [3] proved that there are infinitely many integral isosceles triangle-parallelogram and Heron triangle-rhombus pairs with a common area and a common perimeter. By Fermat's method [4, p. 639], we can give a simple proof of the following result, which is a corollary of Theorem 2.1 in [3].…”

Pairs of equiperimeter and equiareal triangles whose sides are perfect squares

“…Ever since the discovery of right-angled triangles with integer sides, there has been considerable interest in finding triangles as well as polygons with certain geometric properties and all of whose sides are given by integers. Several mathematicians have also considered diophantine problems pertaining to a pair of triangles or other geometric objects (see for instance, [2], [4], [6], [7], [8], [10], [13]). Considerable attention has been given to the problem of finding two triangles with the same perimeter and the same area and such that all the sides and the common area of the two triangles are given by integers (see [1], [3], [9], [12]).…”