  A new isoperimetric estimate is proved for embedded closed curves evolving by
curve shortening flow, normalized to have total length $2\pi$. The estimate
bounds the length of any chord from below in terms of the arc length between
its endpoints and elapsed time. Applying the estimate to short segments we
deduce directly that the maximum curvature decays exponentially to 1. This
gives a self-contained proof of Grayson's theorem which does not require the
monotonicity formula or the classification of singularities.
