Question 53
As shown in Fig. 8.1, $\odot O$ is the inscribed circle touching side $BC$ of $\triangle ABC$ at point $M$, and points $D, E$ are on the segments $AB, AC$, respectively, satisfying $DE \parallel BC$; $\odot O_1$ is the inscribed circle of $\triangle ADE$ tangent to side $DE$ at point $N$; $O_1B, DO$ intersect at point $F$, and $O_1C, EO$ intersect at point $G$. Please prove that $MN$ divides segment $FG$ equally. (posed by Bian Hongping)
Fig. 8.1
Fig. 8.1