Prove that the line drawn through the mid point of one side of a triangle, parallel to another side, intersects the third side at its mid point.

ABC is a triangle;

D is the mid point of AB.

To prove that E is the mid point of AC.

Let F be the mid point of the side AC and not the point E.

Join DF

In ABC, D is the mid point of AB and DE  BC

F is the mid point of AC,

by mid point theorem,

DF  BC, also DE  BC

Thus two intersecting lines DE and DF are parallel to the same line BC

This is not possible.

So F can not be the mid point of AC.

Hence, E is the mid point of AC.