Question

If $\Delta ABC$ is isosceles with AB = AC and C(O,r) is the incircle of the $\Delta ABC$ touching BC at L, prove that L bisects BC.

Answer 1

Given: ABC is an isosceles triangle.

C(O,r) is the incircle of $\Delta ABC$.

$\therefore$ O is the point of intersection of angle bisector.

(i,e.,) OB bisects B and OC bisects C

In triangle ABC,

$AB=BC$ (GIven)

$\Rightarrow \angle C=\angle B$ (Since two sides are equal angle between them also equal)

$\Rightarrow \Delta OCL=\Delta OBL$ (OB bisects triangle(B) and OC bisects triangle(C))

In $\Delta OCL$ and $\Delta OBL$,

$\Delta OLB$ = $\Delta OLC$

$\Delta OBL$ = $\Delta OCL$

$BL=LC$

Thus, L bisects the side BC