KöMaL 2021 B5205

There are four circles given in the plane: circle $k_2$ lies in the interior of circle $k_1$, circle $k_3$ lies in the interior of $k_2$, and circle $k_4$ lies in the interior of $k_3$. Also given are three lines $e_1$, $e_2$ and $e_3$ which are pairwise non-parallel and each line intersects each circle. For all $i=1,2,3$ let the intersections of line $e_i$ with the circles be $A_i$, $B_i$, $C_i$, $D_i$, $E_i$, $F_i$, $G_i$ and $H_i$, in this order. Prove that if $A_1B_1+E_1F_1=C_1D_1+G_1H_1$ and $A_2B_2+E_2F_2=C_2D_2+G_2H_2$ then $A_3B_3+E_3F_3=C_3D_3+G_3H_3$.