Let $(O_a)$ meets $(O'_a)$ at $A_1$, $A_2$; $(O_b)$ meets $(O'_b)$ at $B_1$, $B_2$; $(O_c)$ meets $(O'_c)$ at $C_1$, $C_2$ such that $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ lie on a circle. Let $(O_a)$ meets $(O_b)$ at two points $A_b, B_a$; $(O'_a)$ meets $(O'_b)$ at two points $A'_b, B'_a$ then by Bundle theorem we have quadruple $A_b, B_a, A'_b, B'_a$ are concyclic; Define similarly we have two quadruples $\{B_c, C_b, B'_c, C'_b\}$ and $\{C_a, A_c, C'_a, C'_b\}$ are concyclic. I am looking for a proof that three circles $(A_bB_aA'_bB'_a)$, $(B_cC_bB'_cC'_b)$, $(C_aA_cC'_aC'_b)$ are coaxal.

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