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Prove concyclic and tangency

Source: Japan Olympiad Finals 2014, #4

May 17, 2014
geometrycircumcirclegeometry proposed

Problem Statement

Let Γ \Gamma be the circumcircle of triangle ABCABC, and let ll be the tangent line of Γ\Gamma passing AA. Let D,E D, E be the points each on side AB,ACAB, AC such that BD:DA=AE:EC BD : DA= AE : EC . Line DE DE meets Γ\Gamma at points F,G F, G . The line parallel to ACAC passing D D meets ll at HH, the line parallel to ABAB passing EE meets ll at II. Prove that there exists a circle passing four points F,G,H,I F, G, H, I and tangent to line BC BC.
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