**In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see figure). Show that the line segments AF and EC trisect the diagonal BD.**

**Solution:**

It is given that ABCD is a parallelogram.

and ** (1)**

**(Opposite sides of parallelogram are parallel and equal)**

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

**(2)**

And,

**(F and E are mid-points of CD and AB respectively)** **(3)**

From **(2)** and **(3), **we can say that

AECF is a parallelogram. **(A quadrilateral is a parallelogram if one pair of opposite sides is equal and parallel)**

In , we have

** (AECF is a parallelogram) (4)**

And, F is the mid-point of DC **(Given)** ** (5)**

From **(4)** and **(5)**, we can say that by converse of mid-point theorem, P is the mid-point of DQ.

**(The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.)**

Hence, DP=PQ. ** (6)**

In , we have

** (AECF is a parallelogram) (7)**

And, E is the mid-point of AB ** (Given)** ** (8)**

From **(7)** and **(8)**, we can say that by converse of mid-point theorem, Q is the mid-point of BP.

**(The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.)**

Hence, BQ=PQ ** (9)**

From **(6)** and **(9)**, we get which means that AF and EC trisect the diagonal BD.