That computation is probably something you were already familiar with in disguise: it's just the law of cosines [1][2] along with the fact that a matrix of inner products can be computed with a matrix-matrix multiplication [3].

The first claim is just that for vectors x and y in an inner-product space ||x-y||^2 = <x-y,x-y> = ||x||^2 + ||y||^2 - 2<x,y>, and the second claim is just that if you collect a bunch of vectors into the columns of X and another bunch into the columns of Y, then the (i,j) element of X'Y is <x_i,y_j>, i.e. the inner product of the i'th vector in X with the j'th vector in Y. (Bonus fact: you can use that relationship to translate the statement "this set of pairwise distances can be embedded in a Euclidean space" to an equivalent condition that some simple matrix is negative semidefinite on a subspace orthogonal to the all-ones vector. Euclidean metric embedding is very useful!)

[1] http://en.wikipedia.org/wiki/Law_of_cosines#Vector_formulati...

[2] http://en.wikipedia.org/wiki/Polarization_identity

[3] http://en.wikipedia.org/wiki/Gramian_matrix