CGMO 2011 Problem 8

The $A$ -excircle $(O)$ of $\triangle ABC$ touches $BC$ at $M$ . The points $D,E$ lie on the sides $AB,AC$ respectively such that $DE\parallel BC$ . The incircle $(O_1)$ of $\triangle ADE$ touches $DE$ at $N$ . If $BO_1\cap DO=F$ and $CO_1\cap EO=G$ , prove that the midpoint of $FG$ lies on $MN$ .

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