How to assign coordinates to the elements of a flat metric space Consider the metric space $(M, d \,)$ where set $M$ contains sufficiently many (at least five) distinct elements,
and consider the assignment $c_f$ of coordinates to (the elements of) set $M$,
$c_f \, : \, M \leftrightarrow {\mathbb{R}}^3; \, c_f[ P ] := \{ x_P, y_P, z_P \}$
such that distinct coordinates values are assigned to distinct elements of set $M$, and
such that for the function
$f \, : \, ({\mathbb{R}}^3 \times {\mathbb{R}}^3) \rightarrow {\mathbb{R}};$
$f[ \{ x_P, y_P, z_P \}, \{ x_Q, y_Q, z_Q \} ] :=  $
${\sqrt{ (x_Q - x_P)^2 + (y_Q - y_P)^2 + (z_Q - z_P)^2 }} \equiv {\sqrt{ \sum_{ k \in \{ x \, y \, z \} } (k_Q - k_P)^2 }}$
and for any three distinct elements $A$, $B$, and $J$ $\in M$ holds
$f[ c_f[ A ], c_f[ J ] ] \, d[ B, J ] = f[ c_f[ B ], c_f[ J ] ] \, d[ A, J ]$.
Is the metric space $(M, d \,)$ therefore flat?
(i.e. in the sense of vanishing Cayley-Menger determinants of distance ratios between any five elements of set $M$.)
 A: Yes -- if the coordinates (real number triples) $c_f$ can be assigend to the elements of set $M$ as required in the statement of the question, given the distances (ratios) $d$ and function $f$ as described above then the metric space $(M, d \,)$ is flat.
Because: for any fifteen (real) numbers, $\{ x_\alpha, y_\alpha, z_\alpha \}$, $\{ x_\beta, y_\beta, z_\beta \}$, $\{ x_\gamma, y_\gamma, z_\gamma \}$, $\{ x_\phi, y_\phi, z_\phi \}$ and $\{ x_\lambda, y_\lambda, z_\lambda \}$ the following determinant vanishes  
0 = $  \begin{array}{|cccccc|}
  0 & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\alpha - k_\beta)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\alpha - k_\gamma)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\alpha - k_\phi)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\alpha - k_\lambda)^2}} & 1 & \\
  {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\beta - k_\alpha)^2}} & 0 & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\beta - k_\gamma)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\beta - k_\phi)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\beta - k_\lambda)^2}} & 1 & \\
  {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\gamma - k_\alpha)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\gamma - k_\beta)^2}} & 0 & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\gamma - k_\phi)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\gamma - k_\lambda)^2}} & 1 & \\
  {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\phi - k_\alpha)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\phi - k_\beta)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\phi - k_\gamma)^2}} & 0 & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\phi - k_\lambda)^2}} & 1 & \\
  {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\lambda - k_\alpha)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\lambda - k_\beta)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\lambda - k_\gamma)^2}} & {\small{\sum\limits_{ k \in \{ x \, y \, z \} } (k_\lambda - k_\phi)^2}} & 0 & 1 & \\
  1 & 1 & 1 & 1 & 1 & 0 & \end{array}$.
Consequently, for any five distinct elements $A$, $B$, $J$, $K$ and $Q$ $\in M$ holds
0 = $ \begin{array}{|cccccc|}
  0 & \left(\frac{f[ c_f[ A ], c_f[ B ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ A ], c_f[ J ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ A ], c_f[ K ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ A ], c_f[ Q ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 1 & \\
  \left(\frac{f[ c_f[ B ], c_f[ A ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 0 & \left(\frac{f[ c_f[ B ], c_f[ J ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ B ], c_f[ K ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ B ], c_f[ Q ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 1 & \\
  \left(\frac{f[ c_f[ J ], c_f[ A ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ J ], c_f[ B ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 0 & \left(\frac{f[ c_f[ J ], c_f[ K ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ J ], c_f[ Q ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 1 & \\
  \left(\frac{f[ c_f[ K ], c_f[ A ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ K ], c_f[ B ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ K ], c_f[ J ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 0 & \left(\frac{f[ c_f[ K ], c_f[ Q ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 1 & \\
  \left(\frac{f[ c_f[ Q ], c_f[ A ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ Q ], c_f[ B ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ Q ], c_f[ J ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & \left(\frac{f[ c_f[ Q ], c_f[ K ] ]}{f[ c_f[ A ], c_f[ B ] ]}\right)^2 & 0 & 1 & \\
  1 & 1 & 1 & 1 & 1 & 0 & \end{array}$;
and therefore also
0 = $ \begin{array}{|cccccc|}
  0 & \left(\frac{d[ A, B ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ A, J ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ A, K ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ A, Q ]}{d[ A, B ]}\right)^2 & 1 & \\
  \left(\frac{d[ B, A ]}{d[ A, B ]}\right)^2 & 0 & \left(\frac{d[ B, J ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ B, K ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ B, Q ]}{d[ A, B ]}\right)^2 & 1 & \\
  \left(\frac{d[ J, A ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ J, B ]}{d[ A, B ]}\right)^2 & 0 & \left(\frac{d[ J, K ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ J, Q ]}{d[ A, B ]}\right)^2 & 1 & \\
  \left(\frac{d[ K, A ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ K, B ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ K, J ]}{d[ A, B ]}\right)^2 & 0 & \left(\frac{d[ K, Q ]}{d[ A, B ]}\right)^2 & 1 & \\
  \left(\frac{d[ Q, A ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ Q, B ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ Q, J ]}{d[ A, B ]}\right)^2 & \left(\frac{d[ Q, K ]}{d[ A, B ]}\right)^2 & 0 & 1 & \\
  1 & 1 & 1 & 1 & 1 & 0 & \end{array}$.
Thus, the (normalized) Cayley-Menger determinants of distance ratios between any five elements of set $M$ vanishes; the metric space $(M, d \,)$ is flat. (However, the metric space $(M, d \,)$ is then still not necessarily plane, or even straight.)
The suitable assignment of real number triples $c_f$ to elements of any flat metric space, together with the described function $f$ therefore provides a good (scaled-isometric) representation of the given flat metric space.
