Question 2
Let $AB\Gamma\Delta$ be a quadrilateral inscribed into the circle. With centers $A, B, \Gamma, \Delta$ we draw circles $C_A, C_B, C_\Gamma, C_\Delta$ respectively, not having common points. The circle $C_A$ intersects the sides of the quadrilateral at the points $A_1, A_2$, the circle $C_B$ at the points $B_1, B_2$, the circle $C_\Gamma$ at the points $\Gamma_1, \Gamma_2$ and the circle $C_\Delta$ at the points $\Delta_1, \Delta_2$. Prove that the quadrilateral created by the lines $A_1A_2, B_1B_2, \Gamma_1\Gamma_2$ and $\Delta_1\Delta_2$ is cyclic. (E. Psychas)