  The nature of electronic states due to strong correlation and geometric
frustration on the triangular lattice is investigated in connection to the
unconventional insulating state of Na$_x$CoO$_2$ at $x=0.5$. We study an
extended Hubbard model using a spatially unrestricted Gutzwiller approximation.
We find a new class of charge and spin ordered states at $x=1/3$ and $x=0.5$
where antiferromagnetic (AF) frustration is alleviated via weak charge
inhomogeneity. At $x=0.5$, we show that the $\sqrt{3}a\times2a$ off-plane Na
dopant order induces weak $\sqrt{3}a\times1a$ charge order in the Co layer. The
symmetry breaking enables successive $\sqrt{3}a\times1a$ AF and $2a\times2a$
charge/spin ordering transitions at low temperatures. The Fermi surface is
truncated by the $2a\times2a$ hexagonal zone boundary into small electron and
hole pockets. We study the phase structure and compare to recent experiments.
