If D, E and F are mid-points of the sides BC, CA and AB respectively of a △ABC, prove that AD and FE bisect each other.

△ABC with D, E and F as mid-points of the sides BC, CA and AB is shown below:

D and E are midpoints of BC and CA respectively,

DE = $\dfrac{1}{2}$AB and DE || AB or DE || AF …….(1)

Since,

F is midpoint of AB,

AF = $\dfrac{1}{2}$AB

∴ AF = DE …….(2)

F and D are midpoints of AB and BC respectively,

FD = $\dfrac{1}{2}$AC and FD || AC or FD || AE …….(3)

Since,

E is midpoint of AC,

AE = $\dfrac{1}{2}$AC

∴ FD = AE …….(4)

From 1, 2, 3 and 4 we get,

DE || AF, AF = DE and FD || AE, FD = AE.

Hence, AEDF is a parallelogram.

∴ AD and EF bisect each other.

Hence, AD and EF bisect each other.