KöMaL 2016 A682

Let $A_1B_1C_1D_1E_1F_1A_2B_2C_2D_2E_2F_2$ be a cyclic dodecagon such that its diagonals $A_1A_2$, $B_1B_2$ and $C_1C_2$ are concurrent, the diagonals $A_1A_2$, $E_1E_2$ and $F_1F_2$ are concurrent, the diagonals $C_1C_2$, $D_1D_2$ and $E_1E_2$ are concurrent, and finally the diagonals $B_1B_2$, $D_1D_2$ and $F_1F_2$ are concurrent.Let $k_A$, $k_C$ and $k_E$ be three circular arcs in the interior of the dodecagon's circumcircle such that $k_A$ connects $A_1$ and $A_2$, $k_C$ connects $C_1$ and $C_2$, and $k_E$ connects $E_1$ and $E_2$. Let the intersection points of the three arcs be $B=k_A\cap k_C$, $D=k_C\cap k_E$ and $F=k_A\cap k_E$. Prove that the circular arcs $B_1BB_2$, $D_1DD_2$ and $F_1FF_2$ are concurrent. (These arcs degenerate to line segments if their specified points are collinear.)