KöMaL 2022 B5225

The inscribed circle of triangle $ABC$ is centred at $I$ and has a radius $\varrho$, the radius of the circumscribed circle is $R$. Prove that if $\overline{AI}=R$, then the area of the triangle $ABC$ is $\frac{a\cdot R}{4}+\varrho \cdot a$, where $a$ denotes the length of the side opposite to vertex $A$.