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)
In ΔABC, we have
AB=AC
∴∠ACB=∠ABC (Isosceles triangle theorem)
∴12∠ACB=12∠ABC ..(1)
∴∠OCB=∠OBC and ∠ACO=∠ABO
[OC and OB are bisectors of ∠C and ∠B respectively]
∴OC=OB (Converse of isosceles triangle theorem) ... (2)
(ii)
In ΔABO and ΔACO
AB=AC (Given)
∠ABO=∠ACO ...from (1)
OB=OC ...from (2)
∴ ΔABO≅ΔACO (SAS test of congruence)
∴∠OAB=∠OAC (CPCT)
So,
AO bisects ∠A