D is mid point of side BC of a triangle ABC and E is mid point of BD.If O is mid point of AE , then prove that ar(BOE)=1/8ar(ABC)

IT's easy

join A and D to get AD median

Median divides the triangle into 2 triangles of equal area

therefore, ar(ABD)=1/2 ar(ABC)

now, join A and E  to get AE median 

Similarly we can prove that ar(ABE) = 1/2 ABD

==> ar(ABE)= 1/4 ABC  [ because ar(ABD)= 1/2 ar(ABC) ] ---------1)

Join B and O and we get BO median

Now, ar(BOE)= 1/2 ar(ABE)

==>ar(BOE)= 1/2 x  1/4 ar(ABC)

==>ar(BOE)= 1/8 ar(ABC)