KöMaL 2019 A766

Let $T$ be any triangle such that its side-lengths $a$, $b$, and $c$ and its circumradius $R$ are positive integers. Show that $a)$ the inradius $r$ of $T$ is a positive integer; $b)$ the perimeter $P$ of $T$ is a multiple of four; and $c)$ all three of $a$, $b$, and $c$ are even.