ISL 1994 G3

A circle $C$ has two parallel tangents $L'$ and $L"$ . A circle $C'$ touches $L'$ at $A$ and $C$ at $X$ . A circle $C"$ touches $L"$ at $B$ , $C$ at $Y$ and $C'$ at $Z$ . The lines $AY$ and $BX$ meet at $Q$ . Show that $Q$ is the circumcenter of $XYZ$