We give a simple technique to compute the distance between two points in an n-dimensional Euclidean simplex, where the points are given in barycentric coordinates, using only the edge lengths of that simplex. We then use this technique to verify a few computations which will be used in subsequent papers. The most important application is a formula for intrinsically computing the volume of a Euclidean simplex, which is more efficient (and more natural) than any previously documented methods.