There is a method called Cayley-Menger determinant in order to find if 3 points are collinear, 4 points are coplanar etc. provided that all the pairwise distances are given.
However, in 2-D, there is a much simple way to determine if 3 points, $\{A,B,C\}$ are collinear: Triangle inequality!
$\neg(|AB| + |AC| = |BC|) \wedge \neg(|AB| + |BC| = |AC|) \wedge \neg(|AC| + |BC| = |AB|)$ $\Leftrightarrow$ $A, B, C$ are not collinear
Is there a similar approach in 3-D?
Sure. Think of the areas of the four Triangles. It could be generallized to higher dimensions in a obvious way.