KöMaL 2014 B4649

Let $e_1,e_2,\dots,e_n$ denote different lines on the plane, and let $f$ be a line that is not parallel to any of them. Consider all lines $f_{\alpha}$ parallel to $f$ . Let $S_{\alpha}$ be the centre of mass of the points $f_{\alpha}\cap e_1, f_{\alpha}\cap e_2, \ldots, f_{\alpha}\cap e_n$. Show that the points $S_{\alpha}$ are collinear.