The diagonal SQ of the parallelogram PQRS is divided into three equal parts at K and L . PK intersect SR at M and RL intersect PQ at N . Prove that PMRN is a parallelogram.

Prove that two diagonals of a parallelogram divides it into four triangles of equal areas.

PQRS is cyclic quadrilateral. Extended PQ and SR intersect each other at A . Prove that AP.AQ=AR.AS .

The diagonals AC and BD of the parallelogram ABCD intersect each other at O . Any straight line passing through O intesects the sides AB and CD at the points P and Q respectively . Prove that OP = OQ.

XY is a straight line parallel to the side MT of the parallelogram MNOT , which intersects the side MN at X and the side TO at Y.E and F are two points on XY . If ME and TF are extended, they intersect at P and the extended NE and OF intersect each other at Q . Prove that PQ||MN||TO .

A straight line drawn through D of the parallelogram ABCD intersects AB and the extended part of CB at the points E and F respectively. Prove that AD:AE=CF:CD .

A farmer has a field in the form of a parallelogram PQRS as shown in the figure. He took the mid- point A on RS and joined it to points P and Q. In how many parts of field is divided? What are the shapes of these parts ? The farmer wants to sow groundnuts which are equal to the sum of pulses and paddy. How should he sow? State reasons?

Kamala have drawn a trapezium PQRS of which PQ||SR . If the diagonals PR and QS intersect each other at O , then prove that OP:OR=OQ:OS , If SR=2PQ , then prove that O is a point of trisection of both the diagonals.

ABCD is a parallelogram. AC and BD are the diagonals intersect at O. P and Q are the points of tri section of the diagonal BD. Prove that CQ" ||" AP and also AC bisects PQ (see figure).

Prove that if the lengths of two diagonals of any parallelogram be equal and if the diagonals intersect each other orthogonally, then the parallelogram is a square.

ABCD is a parallelogram. L is a point on BC which divides BC in the ratio 1:2. AL intersects BD at P. M is a point on DC which divides DC in the ratio 1:2 and AM intersects BD in Q. PQ:DB is equal to -