  Given any non-central interior point $o$ of the unit disc $D$, the diameter
$L$ through $o$ is the union of two linear arcs emanating from $o$ which meet
$\partial D$ orthogonally, the shorter of them stable and the longer unstable
(under these boundary conditions). In each of the two half discs bounded by
$L$, we construct a convex eternal solution to curve shortening flow which
fixes $o$ and meets $\partial D$ orthogonally, and evolves out of the unstable
critical arc at $t=-\infty$ and into the stable one at $t=+\infty$. We then
prove that these two (congruent) solutions are the only non-flat convex ancient
solutions to the curve shortening flow satisfying the specified boundary
conditions. We obtain analogous conclusions in the "degenerate" case
$o\in\partial D$ as well, although in this case the solution contracts to the
point $o$ at a finite time with asymptotic shape that of a half Grim Reaper,
thus providing an interesting example for which an embedded flow develops a
collapsing singularity.
