Pairs of rational triangles with equal symmetric invariants

Some diophantine problems concerning a pair of rational triangles with a common circumradius

“…We now have a scalene triangle whose sides are given by (3.15) and a right triangle whose sides are given by (3.16) such that the two triangles have a common circumradius (t 2 1 + 1)(t 2 1 − 4t 1 + 5)/2 and a common inradius 2(t 2 1 − 2t 1 − 1). As a numerical example, when t 1 = 4, we get two triangles with sides 40, 68, 84, and 85, 77, 36, with the common circumradius and common inradius being 85/2 and 14, respectively.…”

“…Regarding problems concerning rational triangles with a common circumradius, it has been shown by Lehmer [8, Theorem XI, p. 101] that there exist infinitely many rational triangles with a common circumradius. Further, Andrica and T ¸urcaş [2] have recently proved that there are no pairs consisting of a rational right triangle and a rational isosceles triangle which have the same circumradius and the same inradius or which have the same circumradius and the same perimeter.…”

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]).…”