Integer Radii of Tritangent Circles in Pythagorean Triangles

Establish that for any Pythagorean triangle with integer side lengths satisfying a^2 + b^2 = c^2, the radii of all four tritangent circles—specifically, the incircle and the three excircles tangent to the triangle’s sides—are integers.

Background

In the subsection on tritangent circles, the authors construct the incircle and three excircles of a triangle and examine their radii for the specific case of a 3–4–5 triangle, displaying both numeric and symbolic values. They then generalize this observation to triangles formed by Pythagorean triples.

Based on numerical experimentation, they explicitly state a conjecture asserting integrality of these radii for Pythagorean triangles and suggest that symbolic formulas offer a pathway to proving the conjecture. This frames an explicit, concrete open question about the arithmetic nature of the inradius and exradii in the Pythagorean setting.