Given an acute triangle ABC with AB<AC. The ex-circles of triangle ABC opposite B and C are centered on B1 and C1, respectively. Let D be the midpoint of B1C1. Suppose that E is the point of intersection of AB and CD, and F is the point of intersection of AC and BD. If EF intersects BC at point G, prove that AG is the bisector of ∠BAC.
geometryangle bisectorexcenter