IMO Shortlist 2002 problem G3

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The circle S has centre O, and BC is a diameter of S. Let A be a point of S such that \angle AOB<120{{}^\circ}. Let D be the midpoint of the arc AB which does not contain C. The line through O parallel to DA meets the line AC at I. The perpendicular bisector of OA meets S at E and at F. Prove that I is the incentre of the triangle CEF.
Source: Međunarodna matematička olimpijada, shortlist 2002