Question 10
G4 (UKR). 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 $B$) such 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 $CC_1$ form a triangle, and the lines $AA_2$, $BB_2$, and $CC_2$ form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.
$$\angle AA_1A_2=\angle AA_2A_1=\angle BB_1B_2=\angle BB_2B_1=\angle CC_1C_2=\angle CC_2C_1.$$
The lines $AA_1$, $BB_1$, and $CC_1$ form a triangle, and the lines $AA_2$, $BB_2$, and $CC_2$ form a second triangle. Prove that all six vertices of these two triangles lie on a single circle.