Question

if the diagonals AC and BD of a quadrilateral ABCD intersect at O such that AO.OD=OB.OC
then show that the quadrilateral is parallelogram

Answer 1

Given : ABCD is a quadrilateral. O is a point inside the quadrilateral ABCD.

To prove : OA + OB + OC + OD > AC + BD

Construction :  Join OA, OB, OC and OD. Also, join AC and BD

Proof :  By triangle in equality the sum of any two sides of a triangle is greater than the third side.

In ΔBOD,

OB + OD > BD …........(1)

Similarly

In ΔAOC,

OA + OC > AC ….........(2)

Adding (1) and (2), we obtain

OB + OD + OA + OC > BD + AC

∴ OA + OB + OC + OD > AC + BD