ABC is a triangle in which AB=AC. If the bisectors of ∠B and ∠C meet AC and AB in D and E respectively, prove that BD=CE.

Question

ABC is a triangle in which AB=AC. If the bisectors of &#x2220; B and &#x2220; C meet AC and AB in D and E respectively, prove that BD=CE.

Philoid · Accepted Answer

Given: AB=AC and BD and AB are angle bisectors of ∠B and ∠C

To prove: BD = CE

Proof:

In ∆ABD and ∆ACE,

∠ABD = ∠B

And ∠ACE = ∠C

But ∠B = ∠C as AB = AC … As in isosceles triangle, base angles are equal

∠ABD = ∠ACE

AB = AC

∠A = ∠A

Thus by ASA property of congruence,

∆ABD ≅ ∆ACE

Hence, we know that, corresponding parts of the congruent triangles are equal

∴ BD = CE