Question 12
In a circle $\Gamma$, a chord $PQ$ is considered such that the segment that joins the midpoint of the smallest arc $\widehat{PQ}$ and the midpoint of the segment $PQ$ measures $1$. Let $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ be three circles tangent to the chord $PQ$ that are in the same half-plane as the center of $\Gamma$ with respect to the straight line $PQ$. Furthermore, $\Gamma_1$ and $\Gamma_3$ are interior tangents to $\Gamma$ and exterior tangents to $\Gamma_2$, and the centers of $\Gamma_1$ and $\Gamma_3$ are in different half-planes with respect to the line determined by the centers of $\Gamma$ and $\Gamma_2$. If the sum of the radii of $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$ is equal to the radius of $\Gamma$, calculate the radius of $\Gamma_2$.