ISL 1999 G5

Let $ABC$ be a triangle, $\Omega$ its incircle and $\Omega_{a}, \Omega_{b}, \Omega_{c}$ three circles orthogonal to $\Omega$ passing through $(B,C),(A,C)$ and $(A,B)$ respectively. The circles $\Omega_{a}$ and $\Omega_{b}$ meet again in $C'$ ; in the same way we obtain the points $B'$ and $A'$ . Prove that the radius of the circumcircle of $A'B'C'$ is half the radius of $\Omega$ .

Attachments: