JBMO-SL 2001 Problem 10

A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$ . Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$ , and both of the rays $(AB$ and $(AC$ . The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $\alpha + \beta + \gamma =1$ .