Problem

Source: 2015 China TST 2 Problem 3

Tags: geometry proposed, circumcircle, TST, tangent, geometry



Let $ \triangle ABC $ be an acute triangle with circumcenter $ O $ and centroid $ G .$ Let $ D $ be the midpoint of $ BC $ and $ E\in \odot (BC) $ be a point inside $ \triangle ABC $ such that $ AE \perp BC . $ Let $ F=EG \cap OD $ and $ K, L $ be the point lie on $ BC $ such that $ FK \parallel OB, FL \parallel OC . $ Let $ M \in AB $ be a point such that $ MK \perp BC $ and $ N \in AC $ be a point such that $ NL \perp BC . $ Let $ \omega $ be a circle tangent to $ OB, OC $ at $ B, C, $ respectively $ . $ Prove that $ \odot (AMN) $ is tangent to $ \omega $


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