  For a bounded simply connected domain $\Omega\subset\mathbb{R}^2$, any point
$z\in\Omega$ and any $0<\alpha<1$, we give a lower bound for the
$\alpha$-dimensional Hausdorff content of the set of points in the boundary of
$\Omega$ which can be joined to $z$ by a John curve with a suitable John
constant depending only on $\alpha$, in terms of the distance of $z$ to
$\partial\Omega$. In fact this set in the boundary contains the intersection
$\partial\Omega_z\cap\partial\Omega$ of the boundary of a John sub-domain
$\Omega_z$ of $\Omega$, centered at $z$, with the boundary of $\Omega$. This
may be understood as a quantitative version of a result of Makarov. This
estimate is then applied to obtain the pointwise version of a weighted Hardy
inequality.
