RO-NMO 2004 Problem 71

On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$ . The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$ . Prove that:

(a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$ ;

(b) $P,A,Q$ are collinear.

Virginia Tica, Vasile Tica