IMO Shortlist 2005 problem G4

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  Avg: 7.0
Let ABCD be a fixed convex quadrilateral with BC=DA and BC not parallel with DA. Let two variable points E and F lie of the sides BC and DA, respectively and satisfy BE=DF. The lines AC and BD meet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R.

Prove that the circumcircles of the triangles PQR, as E and F vary, have a common point other than P.
Source: Međunarodna matematička olimpijada, shortlist 2005