In $Δ$ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects $∠ B A C$

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Solution

In ΔABC, AB = AC
⇒ $∠ B = ∠ C$ [Angles opposite to equal sides are equal]
Also OA and OB are bisectors of angles B and C.

⇒ $∠ O B C = ∠ O C B$
∴ OB = OC [Sides opposite to equal angles are equal]
Now consider, $Δ$ AOB and $Δ$ AOC
OA = OA (Common side)
AB = AC (Given)
OB = OC (Proved)
$Δ A O B ≅ Δ A O C$ [By SSS congruence criterion]
$∴ ∠ O A B = ∠ O A C$
That is OA is bisector ∠A.

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In Δ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that : (i) BO = CO (ii) AO bisects ∠BAC

In $Δ$ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects $∠ B A C$