Let be a triangle, its incircle and three circles orthogonal to passing through and respectively. The circles and meet again in ; in the same way we obtain the points and . Prove that the radius of the circumcircle of is half the radius of .
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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 $\Omega$.
Izvor: Međunarodna matematička olimpijada, shortlist 1999
Školjka 
be a triangle, 
its incircle and 
three circles orthogonal to 
and 
respectively. The circles 
and 
meet again in 
; in the same way we obtain the points 
and 
. Prove that the radius of the circumcircle of 
is half the radius of