KöMaL 2013 B4521

$C$ is an interior point of a segment $AB$ of line $e$. $k_1$, $k_2$ and $k_3$ are semicircles drawn over the line segments $AB$, $AC$ and $CB$ on the same side of line $e$. The midpoints of the arcs of the semicircles are $F_1$, $F_2$ and $F_3$. A circle touches semicircle $k_1$ on the inside at $E$, and also touches the semicircles $k_2$ and $k_3$ on the outside. Show that the midpoint $M$ of the line segment $AB$ and the points $C$, $F_1$, $F_2$, $F_3$ and $E$ are all concyclic.