KöMaL 2005 A373

$P$ is a point in the interior of the quadrilateral $A_1 A_2 A_3 A_4$ that does not lie on either diagonal. The points $B_i$ lie in the interior of each line segment $A_i P$. Let $C_{ij}$ be the intersection of the lines $A_i B_j$ and $A_j B_i$ (1$\le i$<$j$ $\le$4). Prove that the line segments $C_{12} C_{34}$, $C_{13} C_{24}$, $C_{14} C_{23}$ are all concurrent.