SHARYGIN 2015 Problem 14

Let $ABC$ be an acute-angled, nonisosceles triangle. Point $A_1, A_2$ are symmetric to the feet of the internal and the external bisectors of angle $A$ wrt the midpoint of $BC$ . Segment $A_1A_2$ is a diameter of a circle $\alpha$ . Circles $\beta$  and $\gamma$ are defined similarly. Prove that these three circles have two common points.