IMO Shortlist 2004 problem G2

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2. travnja 2012.
Let \Gamma be a circle and let d be a line such that \Gamma and d have no common points. Further, let AB be a diameter of the circle \Gamma; assume that this diameter AB is perpendicular to the line d, and the point B is nearer to the line d than the point A. Let C be an arbitrary point on the circle \Gamma, different from the points A and B. Let D be the point of intersection of the lines AC and d. One of the two tangents from the point D to the circle \Gamma touches this circle \Gamma at a point E; hereby, we assume that the points B and E lie in the same halfplane with respect to the line AC. Denote by F the point of intersection of the lines BE and d. Let the line AF intersect the circle \Gamma at a point G, different from A.

Prove that the reflection of the point G in the line AB lies on the line CF.
Izvor: Međunarodna matematička olimpijada, shortlist 2004