For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we
construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets
have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property
that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tilde\Gamma$ can
be connected via a path, all of which is in $\tilde\Gamma$, which has length
bounded by a fixed constant multiple of the Euclidean distance between $x$ and
$y$.