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

ABCD is a cyclic quadrilaterla. Extended AB and DC intersect at P. Prove that PA.PB = PC.PD.

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 .

In the cyclic quadrilateral ABCD , the diagonal BD bisects the diagonal AC . Prove that ABxxAD=CBxxCD .

In the adjoining figure , the diagonals of the cyxlic quadrilateral PQRS intersect each other at the point X in such a way that anglePRS=65^(@) andangleROS=45^(@) . Find the values of angleSQP and angleRSP .

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. The diagonals AC and BD intersect each other at ‘O’. Prove that ar(DeltaAOD) = ar(DeltaBOC) . (Hint: Congruent figures have equal area)

ABCD is a cyclic quadrilateral. If AB = DC, then prove that AC= BD.

Prove that the internal bisector of any angle of a cyclic quadrilateral and the external bisector of its opposite angle intersect each other on the circumference of the circle.

Multiple Choice Questions (MCQ) PQRS is a cyclic quadrilateral and PQ is a diameter of the circle. If anglePSR = 130^@ then angleQPR =

AB and CD are two chords of a cricle. Extended BA and CD intersect at P. Prove that anglePCB = anglePAD .