Consider the Miquel six circle theorem configuration
Let $A, B, C, D$ are concyclic, $A_1, B_1, C_1, D_1$ are concyclic, and $A, A_1, B_1, ,B$; $B, B_1, C_1 C$, $C, C_1, D_1, D$; $D, D_1, A_1, A$ concyclic. Let $P_1, P_2 = (ACC_1) \cap (DBB_1) $ and $P_4, P_5 = (CAA_1) \cap (BDD_1)$. Then show that $P_1, P_2, P_3, P_4$ are concyclic and $(P_1P_2P_3P_4)$$(CC_1D_1D)$ and $(AA_1B_1B)$ are coaxial.
