In an isosceles triangle ABC, with AB = AC, the bisectors of Angle B and Angle C intersect each other at O. Join A to O. Show that: (i) OB = OC (ii) AO bisects Angle A

(i) In ABC, AB = AC [∵ Given]
Hence, ∠ACB = ∠ACB [∵ Angles opposite to equal sides are equal]
⇒ (1/2)∠ACB = (1/2)∠ABC
⇒ ∠ACO = ∠ABO [∵ OB and OC bisect ∠B and ∠C respectively]
In ΔABO and ΔACO,
AB = AC [∵ Given]
∠ABO = ∠ACO [∵ Proved above]
AO = AO [∵ Common]
Hence, Δ ABO ≅ ΔACO [∵ SAS Congruency Rule]
OB = OC [∵ CPCT]
(ii) ΔABO ≅ ΔACO [∵ Proved above]
∠BAO = ∠CAO [∵ CPCT]
Hence, OA bisects angle A.