Heron triangles and touching circles

Abstract

Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.