Show that the segment joining the mid-points of a pair of opposite sides of a parallelogram, divides it into two equal parts.

Construction: Take P and Q the mid points of AB and CD.

In ||gm ABCD, P is the mid-point of AB and Q is the mid-point of DC.
Also, AB|| DC.

AB = DC and AB || DC

Since P and Q midpoint of AB and DC

So, AP || DQ

∴ AP = 1/2 AB and DQ = 1/2 DC

∴ AP = DQ and AP || DQ

∴Quadrilateral APQD is a parallelogram
Similarly, Quadrilateral PBCQ is a parallelogram.

Now parallelogram APQD and PBCQ are on equal bases AP = PB and between two parallels AB and DC.

∴ ar(||gm APQD) = ar(||gm PBCQ)

Hence proved