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isosceles triangle wanted, circles in an isosceles trapezoid related

Source: Czech and Slovak Match 2000 P5

October 1, 2017
isoscelestrapezoidincircleperpendiculargeometry

Problem Statement

Let ABCDABCD be an isosceles trapezoid with bases ABAB and CDCD. The incircle of the triangle BCDBCD touches CDCD at EE. Point FF is chosen on the bisector of the angle DACDAC such that the lines EFEF and CDCD are perpendicular. The circumcircle of the triangle ACFACF intersects the line CDCD again at GG. Prove that the triangle AFGAFG is isosceles.
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