  A triangle with rational sides and rational area is called a rational
triangle. In this paper we consider three problems of finding pairs of rational
triangles which have a common circumradius as well as either a common perimeter
or a common inradius or a common area. While several similar problems
concerning pairs of rational triangles have been considered in the existing
literature, these three problems have not been studied till now. For each of
these problems, we give a parametric solution and we also indicate how
additional parametric solutions of these problems may be obtained.
