KöMaL 2017 B4902

Let $A_1B_1$, $A_2B_2$, $A_3B_3$ and $A_4B_4$ be four parallel line segments of different lengths given in the plane. For any $1\le i<j\le 4$, let $M_{ij}$ denote the intersection of lines $A_iB_j$ and $A_jB_i$. Show that the lines $M_{12}M_{34}$, $M_{13}M_{24}$ and $M_{14}M_{23}$ are either concurrent or parallel.