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.
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.