Primitive Heronian Triangles With Integer Inradius and Exradii

Abstract

It is well known that primitive Pythagorean triangles have integer inradius and exradii. We investigate the generalization to prim- itive Heronian triangles. In particular, we study the special cases of isosceles triangles and trian- gles with sides in arithmetic progression.

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We also give two families of primitive Heronian triangles, one decomposable and one indecomposable, which have in- teger inradii and exradii. When realized as lattice triangles, these two families have incenters and excenters at lattice points as well. Finally we pose two prob- lems for further research.