Question 72
In plane, let be given two fixed circles $ (O_1) $ and $ (O_2) $ touching each other at $ M $, the radius of $ (O_2) $ is greater than that of $ (O_1) $. $ A $ is the point on $ (O_2) $ such that the points $ O_1 $, $ O_2 $, $ A $ are not collinear. Let $ AB $ and $ AC $ be the tangents to $ (O_1) $ with touching points $ B $ and $ C $. The lines $ MB $ and $ MC $ cut again $ (O_2) $ respectively at $ E $ and $ F $. Let $ D $ be the point of intersection of the line $ EF $ and tangent to $ (O_2) $ at $ A $. Prove that $ D $ moves on a fixed line when $ A $ moves on $ (O_2) $ so that the three points $ O_1 $, $ O_2 $, $ A $ are not collinear.