In a triangle ABC, E is the mid-point of median AD. Show that ar (BED) = $\frac{1}{4}$ ar(ABC).

AD is a median of triangle ABC and BE is the median of ΔABD. 

Since AD is the median of ΔABC, so it will divide ΔABC into two equal triangles.

∴ ar (ΔABD) = ar (ΔADC)

Also, ar (ΔABD) = 1/2 ar(ABC)   .....(i)

Now, In ΔABD, BE is the median,

Therefore, BE will divide ΔABD into two equal triangles

ar (ΔBED) = ar (ΔBAE) and ar (ΔBED) = 1/2 ar(ΔABD)

ar (ΔBED) = 1/2 × [1/2 ar(ABC)] (Using equation (i))

∴ ar (ΔBED) = 1/4 ar(ΔABC)