Question

In an isosceles triangle $ABC$, with $AB=AC$, the bisectors of $\angle B$ and $\angle C$ intersect each other $O$. Join $A$ to $O$. Show that:
$OB=OC$

Answer 1

Since,

$AB=AC$..........(1)

OB is the bisector of $\angle B$

So, $\angle ABO=\angle OBC=\frac{1}{2}\angle B$.......(2)

OC is the bisector of $\angle C$

So, $\angle ACO=\angle OCB=\frac{1}{2}\angle C$.......(3)

$\Rightarrow \angle ACB=\angle ABC$ (Angles opposite to equal sides are equal)

$\frac{1}{2}\angle ACB=\frac{1}{2}\angle ABC$

$\angle OCB=\angle OBC$ From (2) and (3)

Hence,

$OB=OC$ (Sides opposite to equal angles are equal)

Hence proved.