**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)

(5 points)