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 of the quadrilateral ABCD intersect each other at O in such a way 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.

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

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

In trapezium ABCD , BC||AD and AD=4cm . The two diagonals AC and BD intersect at the point O in such a way that (AO)/(OC)=(DO)/(OB)=(1)/(2) . Find the length of BC .

Diagonals of a trapezium ABCD with AB || DC, intersect each other at the point ‘O’. If AB = 2CD, find the ratio of areas of triangles AOB and COD.

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)

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 .