X and Y are the mid-points of the opposite sides AB and DC of a parallelogram ABCD. Prove that `squareAXCY` is a parallelogram.

E is the mid-point of side AB and F is the mid point of side DC of parallelogram ABCD . Prove that AEFD is a parallelogram.

Two points X and Y lie on the diagonal BD of a parallelogram ABCD such that DX = BY. Prove that squareAXCY is a parallelogram.

P and Q are the mid-point of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PQRS is a parallelogram.

In the given figure, M and N are the mid-points of the sides DC and AB respectively of the parallelogram ABCD. If the area of parallelogram ABCD is 48cm^(2) , (i) State the area of the triangle BEC. (ii) name the parallelofram which is equal in area to the triangle BEC.

If P\ a n d\ Q are the mid points of the sides AB and CD of a parallelogram ABCD, prove that DP and BQ cut the trisection which also the points of trisection of DP and BQ respectively.

Show that the opposite sides of a parallelogram are equal.

L and M are the mid-points of sides AB and DC respectively of parallelogram ABCD. Prove that segments DL and BM trisect diagonal AC.

D, E and F are the mid-points of the sides BC, CA and AB respectively of triangle ABC. Prove that: BDEF is a parallelogram.

In Figure, X ,Y are the mid-points of opposite sides A B and D C of a parallelogram A B C Ddot A Y and D X are joined intersecting in P ; C X and B Y are joined intersecting in Qdot Show that DXBY is a parallelogram, P X Q Y is parallelogram.

In the adjoining figure, ABCD and PBCQ are parallelogram. Prove that: APQD is a parallelogram