KöMaL 2013 B4559

The interior angle bisectors drawn from vertices $A$, $B$ and $C$ of triangle ABC intersect the circumscribed circle at the points $D$, $E$ and $F$, respectively. The intersections of the sides of triangles DEF and ABC , starting from $A$, towards $B$, are $G$, $H$, $I$, $J$, $K$ and $L$ in this order. Show that the triangles DGL , EHI and FKJ are all similar.