10th math Converse of Thales theorem

Math Adda

If a line divides any two sides of a triangles proportionally then the line is parallel to the third side

Statement: If a line divides any two sides of a triangles proportionally (in same ratio), then the line is parallel to the third side.

Proof: We are given ABC 

AD / BD = AE / CE

We have to prove: DE is parallel to BC

Let DE's not parallel to BC then an another line DE' is parallel to BC.

Now

AD / BD = AE / CE [Given]

And AD / BD = AE' / CE' [Thales Theorem]

Therefore AE / CE = AE' / CE'

i.e. AE / CE + 1 = AE' / CE' + 1

i.e. AE + CE / CE = AE' + CE' / CE' 

i.e. CE = CE'

But this is not possible until E and E' is coincident.

Thus, our assumption is wrong and DE is parallel to BC.