KöMaL 2010 A500

In space, there are given three prolate spheroids $\mathcal{E}_1$, $\mathcal{E}_2$ and $\mathcal{E}_3$ (ellipsoids which can be obtained by rotating ellipses about their major axes), and a plane $\mathcal{S}$ . The three ellipsoids share one of their foci. Suppose that for each $i$=1,2,3, the surfaces $\mathcal{E}_{i+1}$ and $\mathcal{E}_{i+2}$ have exactly two common points with the plane $\mathcal{S}$ , and denote by $\ell_i$ connecting these common points. Show that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are either concurrent or parallel.