KöMaL 2005 B3844

The legs of a right-angled triangle are $a$ and $b$, its hypotenuse is $c$. The radius of the escribed circle drawn to the leg $b$ (i.e. the one touching that leg on the outside and also touching the extensions of the other two sides) is $\varrho_b$. Prove that $b$+$c$=$a$+2$\varrho_b$.