Given : A ∆ABC in which AD is a line where D is a point on BC and E is the midpoint of AD

To prove: ar(∆BEC) = ar(∆ABC)

Proof: In ∆ABD E is the midpoint of side AD

Area of (∆BDE) = Area of (∆ABE)

Hence Area of (∆BDE) = [Area of (∆ABD)] –1

Now, consider ∆ACD in which E is the midpoint of side AD

Area of (∆ECD) = Area of (∆AEC)

Hence Area of (∆ECD) = [Area of (∆ACD)] –2

Now, adding –1 and –2, we get

Area of (∆BDE) + Area of (∆ECD) = [Area of (∆ABD)] + [Area of (∆ACD)]

area(∆BEC) = [area(∆ABD) + area(∆ACD)]

Area(∆BEC) = Area(∆ABC)

Hence proved