The angle $A$ of triangle $ABC$ is a right angle and the sides $BC$, $CA$ and $AB$ are of lengths $a$, $b$ and $c$ respectively. Each side of the triangle is tangent to the circle $S_1$ which is of radius $r$. Show that $2r=b+c-a$.
Each vertex of the triangle lies on the circle $S_2$. The ratio of the area of the region between $S_1$ and the triangle to the area of $S_2$ is denoted by $R$. Show that $\pi R=-(\pi-1)q^2+2\pi q-(\pi+1),$ where $q=\dfrac{b+c}{a}$. Deduce that $R\leq\frac{1}{\pi(\pi-1)}.$