SHARYGIN 2019 Problem 17

Three circles $\omega_1$ , $\omega_2$ , $\omega_3$ are given. Let $A_0$ and $A_1$ be the common points of $\omega_1$ and $\omega_2$ , $B_0$ and $B_1$ be the common points of $\omega_2$ and $\omega_3$ , $C_0$ and $C_1$ be the common points of $\omega_3$ and $\omega_1$ . Let $O_{i,j,k}$ be the circumcenter of triangle $A_iB_jC_k$ . Prove that the four lines of the form $O_{ijk}O_{1 - i,1 - j,1 - k}$ are concurrent or parallel.