Inradius of Pythagorean Triangle is Integer/Proof 2

Proof
Let $\triangle ABC$ be such that $\angle C$ is a right angle.

Let:
 * $A$ be opposite the side $a$


 * $B$ be opposite the side $b$


 * $C$ be opposite the side $c$


 * Inradius of Pythagorean Triangle is Integer.png

Let $I$ denote the incenter of $\triangle ABC$.

From Solutions of Pythagorean Equation, we have that:


 * $c = m^2 + n^2$

for some $m, n \in \Z_{>0}$, and that the other sides are $m^2 - n^2$ and $2 m n$

, let the sides $a$ and $b$ be such that:

We have that:

As $m$ and $n$ are both integers, it follows that $r$ is also an integer.