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

In ΔABC, AD is median. [∵ Given]
Hence, ar(ABD) = ar(ACD)
⇒ ar(ABD) = 1/2 ar(ABC) …(1)
[∵ A median of a triangle divides it into two triangles of equal areas.]
Similarly, in ΔABD, BE is medium. [∵ E is the mid-point of AD]
Hence, ar(BED) = ar (ABE)
⇒ ar(BED) = 1/2 ar(ABD)
⇒ ar (BED) = 1/2[(1/2)ar(ABC)] [∵ ar (ABD) = (1/2)ar(ABC)]
⇒ ar (BED) = 1/4 ar(ABC)