A Heron triangle and a Diophantine equation

Abstract

Given any positive integer n, it is well known that there always exists a triangle with rational sides a, b and c such that the area of the triangle is n. For any pair of primes (p, q) such that $$p \not \equiv 1$$ (mod 8) and $$p^{2}+1=2q$$ , we look into the possibility of the existence of triangles having rational sides with p as the area and $$p^{-1}$$ as $$\tan \frac{\theta }{2}$$ for one of the angles $$\theta $$ . We also discuss the relation of such triangles with the solutions of certain Diophantine equations.