  Associated to any affine space A endowed with a metric structure of arbitrary
signature we consider the space of affine functionals operating on the space of
quadratic functions of A. On this functional space we characterize a symmetric
bilinear form derived from the metric structure in a functorial way. We explore
the geometrical relations of the relevant objects in this new metric space.
Their properties encode all characteristics known in the literature for
euclidean squared distance matrices, Cayley-Menger matrices and determinants,
squared distance coordinate systems, and Lie and M\"obius sphere geometries.
Birthing this form as Cayley-Menger product, it represents a geometrical
foundation unifying results in all these areas, extending them to metric affine
spaces or bundles.
