Some remarks on Heron triangles (Q2714423)

In this paper six interesting propositions on Heron triangles are presented. (A Heron triangle is a triangle with integer sides and area.) In Proposition 1 a pair of incongruent Heron triangles with area \(F_n\cdot F_{n+1}\cdot F_{n+2}\cdot F_{n+3}\cdot F_{n+4}\cdot F_{n+5}\) is constructed (\(F_k\) is the Fibonacci number) and in Proposition 2 a pair of incongruent Heron triangles is presented with the same semiperimeter (the half sum of sides). The question on Heron triangles whose area has prescribed prime factors is solved in Propositions 3 and 4.NEWLINENEWLINEThe last section in this paper deals with Heron triangles with prescribed radius of the inscribed circle and radius of the circumscribed circle. It is shown in Proposition 5 that for given positive integer there exists a Heron triangle whose radius of the inscribed circle is equal to this integer.