Given: A ∆ABC in which D is the mid-point of BC and E is the mid-point of AD.
To prove: ar(∆BED) = 1/4 ar(∆ABC).
Proof : ∵AD is a median of ∆ABC.
∴ ar(∆ABD) = ar(∆ADC) = 1/2 ar(∆ABC) .....(i)
[∴Median of a triangle divides it into two triangles of equal area) = 1/2 ar(∆ABC) Again,
∵ BE is a median of ∆ABD,
∴ ar(∆BEA) = ar(∆BED) = 1/2 ar(∆ABD)
[∴Median of a triangle divides it into two triangles of equal area]
And 1/2 ar(∆ABD) = 1/2 x 1/2 × ar(∆ABC) [From (i)]
∴ ar(∆BED) = 1/4 ar(∆ABC).