Question

BD and CE are bisectors of ∠ B and ∠ C of an isosceles Δ ABC with AB = AC. Prove that BD = CE.

Answer 1

Δ ABC is isosceles with AB = AC and BD and CE are bisectors of ∠ B and ∠ C We have to prove BD = CE.

Since AB = AC 

∠ABC = ∠ACB ……(i) [Angles opposite to equal sides are equal]

Since BD and CE are bisectors of ∠ B and ∠ C 

∠ ABD = ∠ DBC = ∠ BCE = ECA = ∠B/2 = ∠C/2 …(ii) 

Now, Consider Δ EBC = Δ DCB 

∠ EBC = ∠ DCB [From (i)] 

BC = BC [Common side]

∠ BCE = ∠ CBD [From (ii)] 

By ASA congruence criterion, Δ EBC ≅ Δ DCB 

Since corresponding parts of congruent triangles are equal.

CE = BD 

or, BD = CE 

Hence proved.