Converse of Basic Proportionality Theorem

It is the opposite of basic proportionality theorem, which says that if in a given Triangle a straight line divides the two sides of the Triangle in the same ratio then that straight line is parallel to the third side of the Triangle.

Theorem 2:

If a line divides any of the two sides of a triangle in the same ratio, then that line is parallel to the third side.

ABC is a triangle in which DE divides AC and AB in the same ratio. This states that:

AB/DB = AE/EC

Construction: Draw a line DE’ from point D to E’ at AC. Let us consider DE’||BC.

Proof:

To prove: If DE` || BC, then

AB/DB = AE`/E`C

According to the theorem, AB/DB = AE/EC

Then, accordingly, E and E` must be coincident.

This proves that DE || BC

Proved.

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