  We employ a variational method, in the framework of the Thomas-Fermi
approximation, to study the effect of the symmetry energy on the neutron skin
thickness and the symmetry energy coefficients of various neutron rich nuclei.
We concentrate our interest on $^{208}$Pb, $^{124}$Sn, $^{90}$Zr, and
$^{48}$Ca, although the method can be applied in the totality of medium and
heavy neutron rich nuclei. Our approach has the advantage that the isospin
asymmetry function $\alpha(r)$, which is the key quantity to calculate
isovector properties of various nuclei, is directly related with the symmetry
energy as a consequence of the variational principle. Moreover, the Coulomb
interaction is included in a self-consistent way and its effects can be
separated easily from the nucleon-nucleon interaction. We confirm, both
qualitatively and quantitatively, the strong dependence of the symmetry energy
on the various isovector properties for the relevant nuclei, using possible
constraints between the slope and the value of the symmetry energy at the
saturation density.
