Question

In ΔABC, D, E and F are the midpoints of AB, BC and AC.
Show that AE and DF bisect each other.

Answer 1

Since D and F are mid-points of AB and AC, by Mid-point theorem,
BC = 2DF
Now,
BC = BE + EC
DF = DG + GF
But E is the mid-point of BC,
⇒ BE = EC     ....(i)
Also, AG = GE    ....(G is the mid-point of AE)
Consider ΔABE and ΔACE, by mid-point theorem,
BE = 2DG and EC = 2GF
⇒ 2DG = 2GF   ....[From (i)]
⇒ DG = GF
Hence, AE and DF bisect each other.