Let ABC be a triangle with incircle $\gamma$ and circumcircle $\Gamma$. Let $\Omega$ be the circle tangent to rays $AB, AC,$ and to $\Gamma$ externally, and let $A^{\prime}$ be the tangency point of $\Omega$ with $\Gamma$. Let the tangents from $A^{\prime}$ to $\gamma$ intersect $\Gamma$ again at $B^{\prime}$ and $C^{\prime}$. Finally, let $X$ be the tangency point of the chord $B^{\prime}C^{\prime}$ with $\gamma$. Prove that the circumcircle of triangle $BXC$ is tangent to $\gamma$.
This is something I found from geogebra. Is there a proof of it? Thanks in advance.