**ABCD is a trapezium in which , BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F (see figure). Show that F is the mid-point of BC.**

**Solution:**

**(Given) (1)**

**(Given) (2)**

From **(1)** and **(2)**, we have

In , E is the mid-point of AD and ** (Given)**

Therefore, **by converse of mid-point theorem**, O is the mid-point of BD.

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

In , O is the mid-point of BD and . ** (Proved above)**

Therefore, by **converse of mid-point theorem**, F is the mid-point of BC.

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