BG-MO 1992 Problem 5

Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$ . Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$ , and the line $MN$ intersects the line $CD$ at the point $O$ . Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$ . Prove that $CD=PQ$ . (Plamen Koshlukov)