  Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z
: ... : z^d) in d-dimensional complex projective space are given. It was known
that the number of codimension 2 subspaces intersecting all these lines is
always finite; for a generic configuration it is equal to the d^{th} Catalan
number. We prove that for real tangent lines, all these codimension 2 subspaces
are also real, thus confirming a special case of a general conjecture of B. and
M. Shapiro. This is equivalent to the following result:
  If all critical points of a rational function lie on a circle in the Riemann
sphere (for example on the real line), then the function maps this circle into
a circle.
