  For a compact subset K of the plane and a point x, we define the visible part
of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes
the closed line segment joining x to u.)
  In this paper, we use energies to show that if K is a compact connected set
of Hausdorff dimension larger than one, then for (Lebesgue) almost every point
x in the plane, the Hausdorff dimension of K_x is strictly less than the
Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq
{1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension
of those points where the visible set has dimension larger than
s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0.
