In Fig. 16.135, O is the centre of the circle, Bo is the bisector of ∠ABC. Show that AB=AC.
Given that,
BO is the bisector of ∠ABC
To prove: AB = BC
Proof: ∠ABO = ∠CBO (BO bisector of ∠ABC) (i)
OB = OA (Radii)
Therefore,
∠ABO = ∠DAB (Opposite angle to equal sides are equal) (ii)
OB = OC (Radii)
Therefore,
∠CBO = ∠OCB (Opposite angles to equal sides are equal) (iii)
Compare (i), (ii) and (iii)
∠OAB = ∠OCB (iv)
In triangle OAB and OCB, we have
∠OAB = ∠OCB [From (iv)]
∠OBA = ∠OBC (Given)
OB = OB (Common)
By AAS congruence rule
(By c.p.c.t)
Hence, proved