KöMaL 2016 A668

There is given a positive integer $k$ , some distinct points $A_1,A_2,\ldots,A_{2k+1}$ and $O$ in the plane, and a line $\ell$ passing through $O$ . For every $i=1,\ldots,2k+1$, let $B_i$ be the reflection of $A_i$ about $\ell$, and let the lines $OB_i$ and $A_{i+k}A_{i+k+1}$ meet $C_i$. (The indices are considered modulo $2k+1$: $A_{2k+2}=A_1$, $A_{2k+3}=A_2$, ..., and it is assumed that these intersections occur.) Show that if the points $C_1,C_2,\ldots,C_{2k}$ lie on a line then that line passes through $C_{2k+1}$ also.