Inradius and Circumradius of a Right Triangle

Given a right triangle with legs \(a\) and \(b\), inradius \(r\), and circumradius \(R\):
$$\large (a-r) + (b-r) = 2 R \quad\to\quad a + b = 2 ( R + r )$$

Motivated by this question on the Mathematics Stack Exchange.