Question 62
Let $ABC$ be a triangle, $k$ its incircle and $k_a, k_b, k_c$ three circles orthogonal to $k$ passing through $B$ and $C$, $A$ and $C$, and $A$ and $B$ respectively. The circles $k_a, k_b$ 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 $k$.