mathematical physics - 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 $M$.)