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Turkey TST 2010 Q5

Source:

September 1, 2010
geometrycircumcircleanalytic geometryparallelogramgeometry proposed

Problem Statement

For an interior point DD of a triangle ABC,ABC, let ΓD\Gamma_D denote the circle passing through the points A, E, D, FA, \: E, \: D, \: F if these points are concyclic where BD∩AC={E}BD \cap AC=\{E\} and CD∩AB={F}.CD \cap AB=\{F\}. Show that all circles ΓD\Gamma_D pass through a second common point different from AA as DD varies.
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