Question

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 $\angle A$

Answer 1

(i)

In $\Delta ABC$, we have

$AB=AC$

$\therefore \angle ACB=\angle ABC$ (Isosceles triangle theorem)

$\therefore \frac{1}{2}\angle ACB=\frac{1}{2}\angle ABC$ ..(1)

$\therefore \angle OCB=\angle OBC$ and $\angle ACO=\angle ABO$

[OC and OB are bisectors of $\angle C$ and $\angle B$ respectively]

$\therefore OC=OB$ (Converse of isosceles triangle theorem) ... (2)

(ii)

In $\Delta ABO$ and $\Delta ACO$

$AB=AC$ (Given)

$\angle ABO=\angle ACO$ ...from (1)

$OB=OC$ ...from (2)

$\therefore$ $\Delta ABO\cong \Delta ACO$ (SAS test of congruence)

$\therefore \angle OAB=\angle OAC$ (CPCT)

So,

$AO$ bisects $\angle A$