An active line of research in the networking community studies the distance matrix defined by the node-to-node latencies in the Internet and, in particular, provides a number of quite successful distributed approaches that approximately reconstruct these distances from observations. In such algorithms it is feasible to measure distances among only a linear or near-linear number of node pairs; the rest of the distances are simply not available. The most common framework for Internet measurements of this type is a beacon-based approach: one chooses randomly a constant number of nodes ('beacons') in the network, each node measures its distance to all beacons, and one then has access to only these measurements for the remainder of the algorithm. To obtain theoretical insight into these recent Internet measurement studies, [Kleinberg et al. FOCS'04] formulated a concrete distance reconstruction problem, termed "triangulation", where distances from a given node to beacons form a short node label, and the unobserved distances are inferred from these labels using triangle inequality. While several significant results have been obtained in this framework, all these results include a notion of slack: they provide no guarantees for a small fraction of node pairs. Essentially, for any given positive $\epsilon$ and $\delta$, one can reconstruct all but an $\epsilon$-fraction of distances with multiplicative error at most $1+\delta$, using only a constant number of beacons. In this paper we obtain triangulation-style guarantees \emph{for all node pairs}: we reconstruct all distances with multiplicative error at most $1+\delta$, with only a poly-logarithmic load on each participating node. Our guarantees are for growth-constrained metrics, a well-studied family of metrics which have been proposed as a reasonable abstraction of Internet latencies.