2005 IMO Problems/Problem 5

Let $ABCD$ be a fixed convex quadrilateral with $BC = DA$ and $BC \nparallel DA$ . 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 $P$ .

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2005 IMO Problems/Problem 5