X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD. AY and DX are joined to intersect in P. CX and BY are joined to intersect in Q. Show that PXQY is a parallelogram.

Given: X and Y are the mid-points of opposite sides AB and DC of a parallelogram ABCD.
AX = XB and DY = YC
To Prove: PXQY is a parallelogram.
Proof: AB = DC 
.^.. AB =   DC 
XB = DY......(i)
Also AB ïï DC...... (opp. sides of a ïï^gm)
      XB ïï DY.....(ii)
Since in quadrilateral XBYD, XB = DY and XB ïï DY
.^.. XBYD is a parallelogram.
.^.. DX ïï YB ⇒ PX ïï YQ......(iii)
Similarly we can prove, PY ïï XQ ......(iv)
From (iii) and (iv), we get PXQY is a parallelogram.