Question 127
Let $ABC$ be a triangle with $AB \lt AC \lt BC$ and circumcircle $\Gamma_1$ of center $O$. We consider the circle $\Gamma_2$ with center $D$ lying on the circle $\Gamma_1$, and tangent to the line $BC$ at the point $E$ and tangent to the extension of the side $AB$ at point $F$. The circles $\Gamma_1$ and $\Gamma_2$ intersect at the points $K$ and $G$ (the point $K$ lies in the interior of the triangle $BFE$). If the line $KG$ intersects the lines $FE$ and $CD$ at the points $M$ and $N$, respectively, prove that the quadrilateral $BCNM$ is cyclic.