Question 55
Let $ABC$ be a given triangle. Let $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ be circles with radius $\rho$, centers $A'$, $B'$ and $C'$ respectively, and both the legs of angle $\angle BAC$ are tangents to $\Gamma_A$, both legs of angle $\angle ABC$ are tangents to $\Gamma_B$, both legs of angle $\angle BCA$ are tangents to $\Gamma_C$. The circle $\Gamma$ touches each of the circles $\Gamma_A$, $\Gamma_B$ and $\Gamma_C$ in exactly one point such that all three circles are inside of $\Gamma$, or they are all outside of $\Gamma$. Let $O'$, $I$ and $O$ be the center of $\Gamma$, the incenter of triangle $ABC$ and the circumcenter of triangle $ABC$, respectively.
Show that $O'$ lies on the line $IO$.
Show that $O'$ lies on the line $IO$.