KöMaL 2008 B4090

The angle bisector $f$ drawn from vertex $A$ of an acute-angled triangle ABC intersects side $BC$ at $D$ and the circumscribed circle at $E$. The altitude drawn from $C$ intersects $f$ at $M$ and the circumscribed circle at $Q$. The altitude drawn from $B$ intersects $f$ at $N$ and the circumscribed circle at $P$. Prove that $\frac{BD}{DC} = \frac{PE \cdot AN \cdot ME}{QE \cdot CM \cdot NP}.$