  We study the spin-$1/2$ Heisenberg model on the triangular lattice with the
nearest-neighbor $J_1 > 0$, the next-nearest-neighobr $J_2 > 0$ Heisenberg
interactions, and the additional scalar chiral interaction $J_{\chi}(\vec{S}_i
\times \vec{S}_j) \cdot \vec{S}_k$ for the three spins in all the triangles
using large-scale density matrix renormalization group calculation on cylinder
geometry. With increasing $J_2$ ($J_2/J_1 \leq 0.3$) and $J_{\chi}$
($J_{\chi}/J_1 \leq 1.0$) interactions, we establish a quantum phase diagram
with the magnetically ordered $120^{\circ}$ phase, stripe phase, and
non-coplanar tetrahedral phase. In between these magnetic order phases, we find
a chiral spin liquid (CSL) phase, which is identified as a $\nu = 1/2$ bosonic
fractional quantum Hall state with possible spontaneous rotational symmetry
breaking. By switching on the chiral interaction, we find that the previously
identified spin liquid in the $J_1 - J_2$ triangular model ($0.08 \lesssim
J_2/J_1 \lesssim 0.15$) shows a phase transition to the CSL phase at very small
$J_{\chi}$. We also compute spin triplet gap in both spin liquid phases, and
our finite-size results suggest large gap in the odd topological sector but
small or vanishing gap in the even sector. We discuss the implications of our
results to the nature of the spin liquid phases.
