Question

In Fig. AB = AC, BD and CE are the bisectors of ∠ABC and ∠ACB respectively such that BD and CE intersect each other at O. AO produced meets BC at F. Prove that AF is the right bisector of BC.

Answer 1

Given: A ΔABC in which AB = AC.
BD, the bisector of ∠ABC meets CE, the bisector of ∠ACB at O. AO produced meets BC at F.
To prove: AF is the right bisector of BC.
Proof: We have, AB = AC
⇒ A lies on the right bisector of BC    ...(i)
and ∠ABC = ∠ACB
Now, ∠ABC = ∠ACB
⇒ `(1)/(2)`∠ABC = `(1)/(2)`∠ACB
⇒ ∠OBC = ∠OCB
[∵ BD and CE are bisector of ∠B and∠C respectively]
⇒ OB = OC
[∵ Sides opposite to equal angles are equal]
⇒ O lies on the right bisector of BC    ...(iii)
From (i) and (ii), we obtain
⇒ A and O both lie on the right bisector of BC.
⇒ AO is the right bisector of BC
Hence, AF is the right bisector of BC.
Hence proved.