A. 500. In space, there are given three prolate spheroids , and (ellipsoids which can be obtained by rotating ellipses about their major axes), and a plane . The three ellipsoids share one of their foci. Suppose that for each i=1,2,3, the surfaces and have exactly two common points with the plane , and denote by i connecting these common points. Show that the lines 1, 2 and 3 are either concurrent or parallel.
(Based on the idea of Kristóf Kornis, Budapest)
Solution (in Hungarian)