  We consider the rotation set $\rho(F)$ for a lift $F$ of an area preserving
homeomorphism $f: \t^2\to \t^2$, which is homotopic to the identity. The
relationship between this set and the existence of periodic points for $f$ is
least well understood in the case when this set is a line segment. We show that
in this case if a vector $v$ lies in $\rho(F)$ and has both co-ordinates
rational, then there is a periodic point $x\in \t^2$ with the property that
$$\frac{F^q(x_0)-x_0}q = v$$ where $x_0\in \re^2$ is any lift of $x$ and $q$ is
the least period of $x$.

Dynamical Systems