KöMaL 2018 B4958

The sides of a triangle are $a$, $b$, $c$, the radius of the inscribed circle is $r$, and the radius of the circumscribed circle is $R$. Prove that if $a+b+c =\frac{4}{rR} \text{and} \sqrt{ab}+\sqrt{bc}+\sqrt{ca} =6,$ then $R=2r$.