State and Prove Converse of BPT

Answer: The Contrary of Basic Proportionality Theorem is given as, “If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.”

The illustration can be made using the following example:

Consider a triangle ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC.

The claim that the lines DE being parallel to BC proves our theorem.

Assuming the contrary that, let DE be a line that is non-parallel to BC.

Clearly, there must be another line parallel to BC.

Let it be DF, such that it is parallel to BC.

This is possible only when F and E are same. 

So, DF is the same as the line DE itself.

Therefore, DF is parallel to BC.

Hence the converse of BPT is proven.