Question

ABCD is a parallelogram. P is the mid-point of AB and R is the midpoint of DC as shown in the figure. Lines AR and DP intersect at S and lines PC and BR intersect at Q. Show that PQRS is a parallelogram.

Answer 1

As $R$ and $P$ are mid-point of sides $AB$ and $CD$ so, $DP$ and $AR$ are equal and they are intersected at $S$. Hence, $SR$ and $PS$ are equal. Similarly, $QR$ and $PQ$ are equal. Also, it can be seen that $AR,DP,CP,RB$ are equal as $AD$ and $BC$ are equal. So, all the four sides are equal of the parallelogram $PQRS$. Hence, $PQRS$ are equal.