CN-SMO 2021 Problem 2

In $\triangle ABC$ , $AB=AC>BC$ , point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively, $G$ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$ , then $H$ is the incenter of $\triangle GBC$ .

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