ISL 1995 G4

An acute triangle $ABC$ is given. Points $A_1$ and $A_2$ are taken on the side $BC$ (with $A_2$ between $A_1$ and $C$ ), $B_1$ and $B_2$ on the side $AC$ (with $B_2$ between $B_1$ and $A$ ), and $C_1$ and $C_2$ on the side $AB$ (with $C_2$ between $C_1$ and $B$ ) so that

$$\angle AA_1A_2 = \angle AA_2A_1 = \angle BB_1B_2 = \angle BB_2B_1 = \angle CC_1C_2 = \angle CC_2C_1.$$

The lines $AA_1,BB_1,$ and $CC_1$ bound a triangle, and the lines $AA_2,BB_2,$ and $CC_2$ bound a second triangle. Prove that all six vertices of these two triangles lie on a single circle.