The area of the right angled triangle in terms of its circum radius and inradius is

$\begin{array}{l}R=\frac{a}{2}\Rightarrow 2R=a\\ \mathrm{\Delta }=\frac{1}{2}bc\\ \mathrm{\Delta }=r s\\ \frac{1}{2}bc=r \frac{a+b+c}{2}\Rightarrow bc=r(a+b+c)\end{array}$
$$\begin{gathered} \begin{array}{l}r+2R=\frac{bc}{a+b+c}+a \\ =\frac{bc+a^2+ab+cc}{a+b+c}\\ =\frac{2a^2+2ab+2bc+2ca}{2(a+b+c)}\\ =\frac{a^2+b^2+c^2+2ab+2bc+2ca}{2(a+b+c)}\\ =\frac{{(a+b+c)}^2}{2(a+b+c)}=\frac{2s}{2}=s\\ \mathrm{\Delta }=rs=r(r+2R)\end{array} \end{gathered}$$