IMO Shortlist 1970 problem 8

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Dodao/la: arhiva
2. travnja 2012.
M is any point on the side AB of the triangle ABC. r,r_1,r_2 are the radii of the circles inscribed in ABC,AMC,BMC. q is the radius of the circle on the opposite side of AB to C, touching the three sides of AB and the extensions of CA and CB. Similarly, q_1 and q_2. Prove that r_1r_2q=rq_1q_2.
Izvor: Međunarodna matematička olimpijada, shortlist 1970