KöMaL 2022 B5221

In an acute-angled triangle $ABC$, the points of tangency of the inscribed circle on sides $BC$, $CA$, $AB$ are $D$, $E$, and $F$, respectively. The circumscribed circle of the triangle intersects circle $AEF$ at a point $P$ different from $A$, intersects circle $BFD$ at a point $Q$ different from $B$, and intersects circle $CDE$ at a point $R$ different from $C$. Show that the lines $DP$, $EQ$ and $FR$ are concurrent.