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Question

In a triangle ABC, AB = AC and the bisectors of angles B and C intersect at O. Prove that BO = CO and AO is the bisector of angle BAC.
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Solution
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Since the angles opposite to equal sides are equal,

AB=AC

C=B

B2=C2.

Since BO and CO are bisectors of B and C, we also have

ABO=B2 and ACO=C2.

ABO=B2=C2=ACO.

Consider BCO:

OBC=OCB

BO=CO ....... [Sides opposite to equal angles are equal]

Finally, consider triangles ABO and ACO.

BA=CA ...... (given);

BO=CO ...... (proved);

ABO=ACO (proved).

Hence, by S.A.S postulate

ABOACO

BAO=CAOAO bisects A.

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