IMO Shortlist 1999 problem G5

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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