KöMaL 2015 C1283

The longer base, $AB$ , of trapezium $ABCD$ is not greater than three times the base $CD$ . Lines $e$ and $f$ are parallel to the legs $BC$ and $DA$ , respectively, and each of them halves the area of the trapezium. Let $P$ and $Q$ denote the intersections of $AB$ with $e$ and $f$ , respectively, and let $P'$ and $Q'$ denote their intersections with $DC$ . $a)$ Prove that the intersection $M$ of lines $e$ and $f$ lies on the midline of the trapezium. $b)$ Given that the quadrilateral $PQ'P'Q$ is a parallelogram, find the ratio of the area of triangle $MPQ$ to the area of the trapezium $ABCD$ .