In parallelogram $ABCD$,$P$ is the midpoint of $CD$. Then,$\operatorname{ar}(ABCD) : \operatorname{ar}(PBC) = \dots$

In $\Delta ABC$,$P, Q,$ and $R$ are the midpoints of $AB, BC,$ and $CA$ respectively. Prove that $PBQR, PQCR,$ and $PQRA$ all are parallelograms.

In $\Delta ABC$,$AD$ is a median and $E$ is the midpoint of $AD$. $BE$ is extended to intersect $AC$ at $F$. Prove that $AF = \frac{1}{3} AC$.

$E$ and $F$ are respectively the mid-points of the non-parallel sides $AD$ and $BC$ of a trapezium $ABCD$. Prove that $EF \parallel AB$ and $EF = \frac{1}{2}(AB + CD)$.

$ABCD$ is a trapezium in which $AB \parallel DC$,$BD$ is a diagonal,and $E$ is the mid-point of $AD$. $A$ line is drawn through $E$ parallel to $AB$ intersecting $BC$ at $F$. Show that $F$ is the mid-point of $BC$.

$ABCD$ is a quadrilateral in which $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ respectively (see figure). $AC$ is a diagonal. Show that: $PQ = SR$.