KöMaL 2021 B5210

The parabolas $\mathcal{P}_1$, $\mathcal{P}_2$ and $\mathcal{P}_3$ have the same focus, and any two of them intersect at exactly two points. Let $e_{ij}$ denote the line passing through the two intersections of the parabolas $\mathcal{P}_i$ and $\mathcal{P}_j$. Show that the lines $e_{12}$, $e_{13}$ and $e_{23}$ are concurrent.