  An important theorem of Beck says that any point set in the Euclidean plane
is either ``nearly general position'' or ``nearly collinear'': there is a
constant C>0 such that, given n points in the plane with at most r$ of them
collinear, the number of lines induced by the points is at least Cr(n-r).
  Recent work of Gutkin-Rams on billiards orbits requires the following
elaboration of Beck's Theorem to bichromatic point sets: there is a constant
C>0 such that, given n red points and n blue points in the plane with at most r
of them collinear, the number of lines spanning at least one point of each
color is at least Cr(2n-r).
