Question 150
Three circles $\Gamma_{1}$, $\Gamma_{2}$, $\Gamma_{3}$ are pairwise externally tangent, with $\Gamma_{1}$, the smallest, having radius $1$, and $\Gamma_{3}$, the largest, having radius $25$. Let $A$ be the point of tangency of $\Gamma_{1}$ and $\Gamma_{2}$, $B$ be the point of tangency of $\Gamma_{2}$ and $\Gamma_{3}$, and $C$ be the point of tangency of $\Gamma_{1}$ and $\Gamma_{3}$. Suppose now that triangle $ABC$ has circumradius $1$ as well. The radius of $\Gamma_{2}$ can then be written in the form $p/q$, where $p$ and $q$ are relatively prime positive integers. Find the value of the product $pq$.