The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the problem can be formulated as a nuclear norm minimization problem. In this paper, this minimization program is recast as a matrix completion problem of a low-rank $r$ Gram matrix with respect to a suitable basis. The well known restricted isometry property can not be satisfied in this scenario. Instead, a dual basis approach is introduced to theoretically analyze the reconstruction problem. If the Gram matrix satisfies certain coherence conditions with parameter $ν$, the main result shows that the underlying configuration of $n$ points can be recovered with very high probability from $O(nrν\log^{2}(n))$ uniformly random samples. Computationally, simple and fast algorithms are designed to solve the Euclidean distance geometry problem. Numerical tests on different three dimensional data and protein molecules validate effectiveness and efficiency of the proposed algorithms.