# Incircle of a triangle, and circumdcribed circle around the triangle created by the center of the incircle

$O$ is the center of the incircle of $\Delta ABC$. If $AB=6$ and $\angle ACB=120^{\circ}$, what is the radius of the circumscribed circle of $\Delta AOB$?

• Aren't there $3$ excircles to a triangle? – Shraddheya Shendre Nov 25 '16 at 16:25
• by excircle I mean a circle surrounding the triangle – Irena Alexieva Nov 25 '16 at 16:32
• That's a circumcircle. Excircle is a completely different thing. – Shraddheya Shendre Nov 25 '16 at 16:33
• I see the error in my terminology and have fixed it. English is not my first language. – Irena Alexieva Nov 25 '16 at 16:36

Let $\alpha, \beta, \gamma$ be the interior angles of the triangle ABC in the usual way, so we know $\gamma=120°$. We have $\sphericalangle BAO = \frac{\alpha}{2}$ and $\sphericalangle ABO = \frac{\beta}{2}$ (center of incircle is the intersection of angle bisectors), hence (sum of inner angles in triangle ABO)
$$\sphericalangle AOB = 180° -\frac{\alpha+\beta}{2} = 150°,$$
as $\alpha+\beta = 60°$ (sum of inner angles of triangle ABC).
The law of sines in triangle ABO says that $\frac{AB}{\sin \sphericalangle AOB} = 2r$, where $r$ is the radius of the circumscribed circle of ABO. We get $r=6$.