Question

O is the centre of the incircle for $△$ABC. Find $\angle$BOC if $\angle$BAC = 40${}^o$.

Answer 1

Incentre is defined as Intersection of angle bisectors of vertices of a triangle.

=> AF, BE, CD are angle bisectors.

=> $\angle BAF=\angle FAC=20$

=>Let $\angle ABE=\angle FBE=x$

=>Let $\angle BCD=\angle ACD=y$

Now, using the exterior angle property of a triangle, $\angle FOC=20+y$ and $\angle BOF=20+x$

in triangle BOC, using the property that sum of all angles of triangle is $180$, we can say that

$20+y+20+x+x+y=180$

=> $x+y=70$

$\angle BOC=20+y+20+x=110$