Question 19
Let $ABC$ be a non-isosceles triangle and $I$ the centre of its incircle. Let $D$ be a point on $[BC]$ (other than $B$ and $C$). The circumcircle of $DIB$ meets $(AB)$ a second time at $E \neq B$, and the circumcircle of $DIC$ meets $(AC)$ a second time at $F \neq C$. The circumcircle of $DEF$ meets $(AB)$ and $(AC)$ a second time at $M$ and $N$, respectively. The lines $(IB)$ and $(DE)$ meet at $P$ and the lines $(IC)$ and $(DF)$ meet at $Q$. Show that the lines $(EN)$, $(FM)$ and $(PQ)$ are parallel.