The diagonals of the square ABCD intersect each other at O .Let `OPbotAB` is constructed. Prove that `angleAOP =anglePBO`

The diagonals of the quadrilateral ABCD intersect each other at O in such a way that (AO)/(BO)=(CO)/(DO) . Prove that ABCD is a trapezium.

Two chords AB and CD of the circle with centre at O intersect each other at an internal point P of the centre. Prove that angle AOD+ angle BOC=2 angle BPC

The diagonals of a quadrilateral ABCD intersect each other at point ‘O’ such that (AO)/(BO) = (CO)/(DO) . Prove that ABCD is a trapezium

The diagonals AC and BD of a quadrilateral ABCD intersects each other at O such that DeltaAOD=DeltaBOC . Prove that ABCD is trapezium.

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 .

If the diagonals of the quadrilateral ABCD intersect each other orthogonally,then prove that AB^(2) + CD^(2) = BC^(2) + DA^(2)

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.

The medians BE and CF of the DeltaABC intersect each other at G and the line segment EF intersects AG at O. Prove that OA = 3.OG.

Two chords AB and CD of a circle with centre O intersect each other at the point P. prove that AOD+ angle BOC = 2 angle PBC . If angle BOC are supplementary to each other, then prove that the two chords are perpendicular to each other.

In the figure, diagonals AC and BD of a trapezium ABCD with AB || DC intersect each other at O. Prove that ar(DeltaAOD) = ar(Delta BOC).