Converse Of Midpoint Theorem

Theorem: The line through the midpoint of one side of a triangle when drawn parallel to a second side bisects the third side.

Prerequisites:
Midpoint theorem (proof)
Unique parallel through a point (proof)

Proof:

Let ABC be a triangle such that D is the midpoint of the line AB. Let E be a point on the line AC, such that DE $\parallel$ BC.

If possible, let us assume that E is not the midpoint of the line AC. Let E' be the midpoint of the line AC. Let us draw a line DE'.

Now in $\triangle ABC$, DE' is the line joining the midpoints of the lines AB and AC. Therefore, by Midpoint theorem, DE' $\parallel$ BC.

But we have DE $\parallel$ BC. This is a contradiction, as only a unique line can pass through the point D which is parallel to the line BC.

Hence, E is the midpoint of the line AC, i.e., line DE bisects the side AC.

Q.E.D.


Recommended:
Midpoint Theorem
Basic Proportionality Theorem
Properties of centroid