**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

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.

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