NMC 2010 Problem 2

Three circles $\Gamma_{A}, \Gamma_{B}$ and $\Gamma_{C}$ share a common point of intersection $O$. The other common of $\Gamma_{A}$ and $\Gamma_{B}$ is $C$, that of $\Gamma_{A}$ and $\Gamma_{C}$ is $B$ and that of $\Gamma_{C}$ and $\Gamma_{B}$ is $A$. The line $A O$ intersects the circle $\Gamma_{C}$ in the poin $X \neq O$. Similarly, the line $B O$ intersects the circle $\Gamma_{B}$ in the point $Y \neq O$, and the line $C O$ intersects the circle $\Gamma_{C}$ in the point $Z \neq O$. Show that

$$\frac{|A Y||B Z||C X|}{|A Z||B X||C Y|}=1$$