ISL 1997 Problem 25

Let $X,Y,Z$ be the midpoints of the small arcs $BC,CA,AB$ respectively (arcs of the circumcircle of $ABC$ ). $M$ is an arbitrary point on $BC$ , and the parallels through $M$ to the internal bisectors of $\angle B,\angle C$ cut the external bisectors of $\angle C,\angle B$ in $N,P$ respectively. Show that $XM,YN,ZP$ concur.