Question 37
Consider three circles $\Gamma_1, \Gamma_2$ and $\Gamma_3$, with centres $O_1, O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1, \Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$.
Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.
Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.