Subcontests
(4)Prove concyclic and tangency
Let Γ be the circumcircle of triangle ABC, and let l be the tangent line of Γ passing A. Let D,E be the points each on side AB,AC such that BD:DA=AE:EC. Line DE meets Γ at points F,G. The line parallel to AC passing D meets l at H, the line parallel to AB passing E meets l at I. Prove that there exists a circle passing four points F,G,H,I and tangent to line BC.