Points M and N are taken on the diagonal AC of a parallelogram ABCD such that AM= CN. Prove that BMDN is a parallelogram.

E and F are points on diagonal AC of a parallelogram ABCD such that AE=CF. Show that BFDE 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.

M and N are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CN is parallel to AM.

The diagonal BD of a parallelogram ABCD bisects angles B and D. Prove that ABCD is a rhombus.

P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP.

Prove that the diagonals of a parallelogram bisect each other.

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.

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

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

The vector -2hati+4hatj+4hatk and -4hati-4hatk represent the diagonals BD and AC of a parallelogram ABCD. Then, find the area of the parallelogram.