IMO Shortlist 2006 problem G9


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2. travnja 2012.
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Points A_{1}, B_{1}, C_{1} are chosen on the sides BC, CA, AB of a triangle ABC respectively. The circumcircles of triangles AB_{1}C_{1}, BC_{1}A_{1}, CA_{1}B_{1} intersect the circumcircle of triangle ABC again at points A_{2}, B_{2}, C_{2} respectively (A_{2}\neq A, B_{2}\neq B, C_{2}\neq C). Points A_{3}, B_{3}, C_{3} are symmetric to A_{1}, B_{1}, C_{1} with respect to the midpoints of the sides BC, CA, AB respectively. Prove that the triangles A_{2}B_{2}C_{2} and A_{3}B_{3}C_{3} are similar.
Izvor: Međunarodna matematička olimpijada, shortlist 2006