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 \[R\leq\frac{1}{\pi(\pi-1)}.\]
