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A circle, many lines and points - ref. of G in AB lies on CF

Source: ISL 2004; Greek TST 2005; Moldova TST 2005; etc.

January 19, 2005
geometryreflectioncircumcircleIMO Shortlist

Problem Statement

Let Γ\Gamma be a circle and let dd be a line such that Γ\Gamma and dd have no common points. Further, let ABAB be a diameter of the circle Γ\Gamma; assume that this diameter ABAB is perpendicular to the line dd, and the point BB is nearer to the line dd than the point AA. Let CC be an arbitrary point on the circle Γ\Gamma, different from the points AA and BB. Let DD be the point of intersection of the lines ACAC and dd. One of the two tangents from the point DD to the circle Γ\Gamma touches this circle Γ\Gamma at a point EE; hereby, we assume that the points BB and EE lie in the same halfplane with respect to the line ACAC. Denote by FF the point of intersection of the lines BEBE and dd. Let the line AFAF intersect the circle Γ\Gamma at a point GG, different from AA. Prove that the reflection of the point GG in the line ABAB lies on the line CFCF.
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