We introduce a new distance-based phylogeny reconstruction technique which provably achieves, at sufficiently short branch lengths, a logarithmic sequence-length requirement---improving significantly over previous polynomial bounds for distance-based methods and matching existing results for general methods. The technique is based on an averaging procedure that implicitly reconstructs ancestral sequences. In the same token, we extend previous results on phase transitions in phylogeny reconstruction to general time-reversible models. More precisely, we show that in the so-called Kesten-Stigum zone (roughly, a region of the parameter space where ancestral sequences are well approximated by "linear combinations" of the observed sequences) sequences of length $O(\log n)$ suffice for reconstruction when branch lengths are discretized. Here $n$ is the number of extant species. Our results challenge, to some extent, the conventional wisdom that estimates of evolutionary distances alone carry significantly less information about phylogenies than full sequence datasets.