Incircles and Excircles of Pythagorean triangles

This post will reveal the connection between my two previous posts: one on the Star Trek lemma and one on Pythagorean triples. In the process of writing the latter, I looked at the Wikipedia article on Pythagorean triples and noticed this curious paragraph. In every Pythagorean triangle, the radius of the incircle and the radii of the three excircles are positive integers. Specifically, for a primitive triple the radius of the incircle is r = n(m − n), and the radii of the excircles opposite the sides m2 − n2, 2mn, and the hypotenuse m2 + n2 are respectively m(m − n), n(m + n), and m(m + n). The citation for the paragraph above was the book by my former officemate, which led to the post on the Star Trek lemma. The passage in Arthur Baragar’s book that Wikipedia cites is Exercise 15.3. Let ΔABC be a right angle triangle with sides of integer length. Prove that the inradius r and the exradii ra, rb, and rc are all integers. I don’t know whether Arthur discovered this theorem, but I’ll…