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.
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.
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