**D, E and F are respectively the mid-points of the sides BC, CA and AB of a . Show that**

**(i)** BDEF is a parallelogram. **(ii)**

**(iii)**

**Solution (i)**

In , D is the mid-point of BC and E is the mid-point of AC.

Therefore, **by mid-point theorem**, we have

**(A line joining the mid-points of the two sides of a triangle is parallel to the third side.)**

** (1)**

Similarly, **by mid-point theorem**, we have ** (2)**

From **(1)** and **(2)**, we can say that

BDEF is a parallelogram.

**(A quadrilateral is a parallelogram if both the pairs of opposite sides are parallel.)**

**Solution(ii)**

We have already proved above that BDEF is a parallelogram. In the same way, we can prove that FECD and AEDF are parallelograms using **mid-point theorem**.

**(3)**

**(Diagonal of a parallelogram divides it into two triangles of equal areas.)**

And, **(4)**

**(Diagonal of a parallelogram divides it into two triangles of equal areas.)**

And, **(5)**

**(Diagonal of a parallelogram divides it into two triangles of equal areas.)**

We have

Putting **(3)**, **(4) and (5) **in the above equation, we get

**Solution (iii)**

**(6)**

But, we have **(Proved above) (7)**

We also have ** (8)**

**(Parallelograms on the same base and between the same parallels are equal in area.)**

Putting **(7)** and **(8)** in equation **(6)**, we get