Question 51
$I$ is the incenter of the triangle $ABC$. A circle with center $I$ meets the side $BC$ at $D$ and $P$, with $D$ nearer to $B$. Similarly, it meets the side $CA$ at $E$ and $Q$, with $E$ nearer to $C$, and it meets $AB$ at $F$ and $R$, with $F$ nearer to $A$. The lines $EF$ and $QR$ meet at $S$, the lines $FD$ and $RP$ meet at $T$, and the lines $DE$ and $PQ$ meet at $U$. Show that the circumcircles of $DUP$, $ESQ$ and $FTR$ have a single point in common.