IMO Shortlist 2000 problem G1

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Dodao/la: arhiva
2. travnja 2012.
In the plane we are given two circles intersecting at X and Y. Prove that there exist four points with the following property:

(P) For every circle touching the two given circles at A and B, and meeting the line XY at C and D, each of the lines AC, AD, BC, BD passes through one of these points.
Izvor: Međunarodna matematička olimpijada, shortlist 2000