Given : D is the midpoint of side BC of ∆ABC and E is the midpoint of BD and O is the midpoint of AE

To prove : ar(∆BOE) = ar(∆ABC)

Proof : Consider ∆ABC here D is the midpoint of BC

Area of (∆ABD) = Area of (∆ACD)

Area(∆ABD) = Area(∆ABC)—1

Now, consider ∆ABD here E is the midpoint of BD

Area of (∆ABE) = Area of (∆AED)

Area(∆ABE) = Area(∆ABD)—2

Substituting –1 in –2 , we get

Area(∆ABE) = ( Area(∆ABC))

Area(∆ABE) = Area(∆ABC)—3

Now consider ∆ABE here O is the midpoint of AE

Area of (∆BOE) = Area of (∆AOB)

Area(∆BOE) = Area(∆ABE)—4

Now, substitute –3 in –4 , we get

Area(∆BOE) = ( Area(∆ABC))

area(∆BOE) = area(∆ABC)

Hence proved