Question

In a ∆ABC, D and E are points on sides AB and AC respectively such that BD = C’E. If ∠B = ∠C, show that DE || BC.

Answer 1

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.