D is the mid-point of side BC of Δ ABC and E is the mid-point of BD. If O is the mid-point of AE, prove that ar(ΔBOE) = ar(Δ ABC)

Join A and D to get AD median

(Median divides the triangle into two triangles of equal area)

Therefore,

Area (ABD) = Area (ABC)

Now,

Join A and E to get AE median

Similarly,

We can prove that,

Area (ABE) = Area (ABD)

Area (ABE) = ABC (Area (ABD) = Area (ABC) (i)

Join B and O and we get BO median

Now,

Area (BOE) = Area (ABE)

Area (BOE) = * Area (ABC)

Area (BOE) = Area (ABC)