If the diagonals `A C ,B D` of a quadrilateral `A B C D ,` intersect at `O ,` and seqarate the quadrilateral into four triangles of equal area, show that quadrilateral `A B C D` is a parallelogram. GIVEN : A quadrilateral `A B C D` such that its diagonals `A C` and `B D` intersect at `O` and separate it into four parts such that `a r( A O B)=a r( B O C)=a r( C O D)=a r( A O D)` TO PROVE : Quadrilateral `A B C D` is a parallelogram.

Explanation

`ar(AOD)=ar(BOC)->"(Given)"`

`"Add ar"(AOB)" to both the sides "`

`ar(AOD)+ar(AOB)=ar(BOC)+ar(AOB)`

`=>ar(ABD)=ar(ABC)`

`/_\ABD and /_\ABC", lies on the same base AB and have equal area "`

`therefore "AD is parallel to AB"`

`"similarly, DC is parallel to AB"`

`"opposite sides of Quadrilateral ABCD are parallel"`

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