KöMaL 2015 A655

Two circles, $k_1$ and $k_2$ meet at points $A$ and $B$ . Points $C$ and $D$ lie on $k_1$, while points $E$ and $F$ lie on circle $k_2$ in such a way that $A$ , $C$ , $E$ are collinear and $B$ , $D$ , $F$ are collinear, too. Points $G$ and $H$ are other two points on lines $ACE$ and $BDF$ , respectively. The line $CH$ meets $FG$ and $k_1$ the second time at $I$ and $J$ , respectively. The line $DG$ meets $EH$ and $k_1$ the second time at $K$ and $L$ , respectively. Circle $k_2$ meets the lines $EHK$ and $FGI$ the second time at $M$ and $N$ , respectively. The points $A,B,C,\ldots,N$ are distinct. Show that $I$ , $J$ , $K$ , $L$ , $M$ and $N$ are either concyclic or collinear.