Given : ∆ABC in which D and E are points on sides AB and AC respectively, such that BD = CE.
To prove : DE || BC.
Proof: In ∆ABC, we have
∠B = ∠C
⇒ AC = AB
⇒ AB = AC
[Sides opposite equal angles arc equal]
⇒ AD + DB = AE + EC
But BD = CE
⇒ AD = AE
Thus, we have
AD = AE and BD = CE
⇒ \(\frac{AD}{BD} = \frac{AE}{BE}\)
Therefore, by converse of Basic proportionality theorem, we get DE || BC.