Tests for parallelogram

`square` ABCD is a parallelogram, P and Q are midpoints of side AB and DC respectively, then prove `square` APCQ is a parallelogram.

Using opposite angles test for parallelogram, prove that every rectangle is a parallelogram.

In the given figure, G is the point of concurrence of medians of Δ DEF. Take point H on ray DG such that D-G-H and DG = GH, then prove that `square` GEHF is a parallelogram.

Prove that quadrilateral formed by the intersection of angle bisectors of all angles of a parallelogram is a rectangle.(shown in the given figure)

In the given figure, if points P, Q, R, S are on the sides of parallelogram such that AP = BQ = CR = DS then prove that `square` PQRS is a parallelogram.

Ratio of two adjacent sides of a parallelogram is 3 : 4, and its perimeter is 112 cm. Find the length of its each side.