KöMaL 2011 B4369

Each of the circles $k_1$, $k_2$ and $k_3$ passes through a point $P$, and the circles $k_i$ and $k_j$ also pass through the point $M_{i,j}$. Let $A$ be an arbitrary point of circle $k_1$. Let $k_4$ be an arbitrary circle passing through $A$ and $M_{1,2}$, and let $k_5$ be an arbitrary circle passing through $A$ and $M_{1,3}$. Show that if the other intersections of the pairs of circles $k_4$ and $k_2$, $k_5$ and $k_3$, $k_4$ and $k_5$ are $B$, $C$ and $D$, respectively, then the points $M_{2,3}$, $B$, $C$, $D$ are either concyclic or collinear.