  We investigate the magnetic properties and Mott transition in the Hubbard
model for weakly coupled chains on the anisotropic triangular lattice. Taking
into account 120$^\circ$ N\'eel, and collinear orderings, the magnetic phase
diagram is studied at zero temperature and half-filling by the variational
cluster approximation. We found that when the on-site Coulomb repulsion $U$ is
relatively large, nonmagnetic insulator, which is a candidate of the spin
liquid state, is realized for wide range of the interchain hopping from quasi
two-dimensional to almost one-dimensional regime. When the interchain hopping
is relatively large, this nonmagnetic insulator becomes magnetic states as $U$
decreases. For rather small interchain hopping, it changes to the paramagnetic
metal, thus purely paramagnetic metal-insulator transition (Mott transition)
takes place. Implications of our results for the Cs$_2$CuBr$_4$ and
Cs$_2$CuCl$_4$ are discussed.
