KöMaL 2011 A529

There is given a circle $k$ on the plane, a chord $AB$ of $k$, furthermore four interior points, $C$, $D$, $E$ and $F$, on the line segment $AB$. Draw an arbitrary chord $X_1 X_2$ of $k$ through point $C$, a chord $Y_1 Y_2$ through $D$, a chord $U_1 U_2$ through $E$, finally a chord $V_1 V_2$ through $F$ in such a way that $X_1$, $Y_1$, $U_1$ and $V_1$ lie on the same side of the line $AB$, and $\frac{AX_1\cdot BX_2}{X_1X_2} =\frac{AY_2\cdot BY_1}{Y_1Y_2} =\frac{AU_1\cdot BU_2}{U_1U_2} =\frac{AV_2\cdot BV_1}{V_1V_2}$ holds. Let $Z$ be the intersection of the lines $X_1 X_2$ and $Y_1 Y_2$, and let $W$be the intersection of $U_1 U_2$ and $V_1 V_2$. Show that the lines $ZW$ obtained in this way are concurrent or they are parallel to each other.