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

(i)InΔABC, we haveAB=AC∴∠ACB=∠ABC(Isosceles triangle theorem)∴12∠ACB=12∠ABC. (1)∴∠OCB=∠OBCand∠ACO=∠ABO[OC and OB are bisectors of∠Cand∠Brespectively]∴OC=OB(Converse of isosceles triangle theorem) ... (2)(ii)InΔABOandΔACOAB=AC(Given)∠ABO=∠ACO...from (1)OB=OC...from (2)∴ΔABO≅ΔACO(SAS test of congruence)∴∠OAB=∠OAC(CPCT)So,AObisects∠A