We present an algorithm based on local generic position (LGP) to isolate the complex or real roots and their multiplicities of a zero-dimensional triangular polynomial system. The Boolean complexity of the algorithm for computing the real roots is single exponential: $\tilde{\mathcal {O}}_B(N^{n^2})$, where $N=\max\{d,\tau\}$, $d$ and $\tau$, is the degree and the maximum coefficient bitsize of the polynomials, respectively, and $n$ is the number of variables.