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If the diagonals A C ,\ B D of a q...

If the diagonals `A C ,\ B D` of a quadrilateral `A B C D ,` intersect at `O ,` and separate the quadrilateral into four triangles of equal area, show that quadrilateral `A B C D` is a parallelogram.

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Given A quad. ABCD whose diagonals AC and BD intersect at O in such a way that

`ar(triangleAOB)=ar(triangleBOC)=ar(triangleAOD)=ar(triangleCOD)`.
TO PROVE ABCD is a ||gm.
PROOF `ar(triangleAOD)=ar(triangleBOC)`
`rArr ar(triangleAOD)+ar(triangleAOB)=ar(triangleBOC)+ar(triangleAOB)`
`rArr ar (triangleABD)=ar(triangleABC)`.
Also, `triangleABD and triangle ABC` have the same base.
`therefore` they lie between the same parallels AB and DC, i.e., AB ||DC
[THeorem 9].
Similarly, AD||BC.
Hence, ABCD is a parallelogram.
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