In an acute-angled triangle ${ABC}$ , its incenter $I$ and circumcenter $O$ are marked. The lines ${AI}$ and ${CI}$ have second intersections with the circumcircle of ${ABC}$ at points $N$ and $M$ respectively. The segments ${MN}$ and ${BO}$ intersect at the point $X$ . Prove that the lines ${XI}$ and ${AC}$ are perpendicular. Fedor Ivlev
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