Question:
is there a theorem that guarantees that
$\mathcal{P}\subset\mathbb{E}^n$ is finite set of points in a Euclidean space and all radii of the $(n-1)$-spheres that are defined by the $n$-simplices with $n$ corners in $\mathcal{P}$ are equal $\implies$
the centers of the circumspheres are identical.

Calculating the radius of a simplexes circumsphere is possible by means of sidelengths , cf e.g. https://westy31.home.xs4all.nl/Circumsphere/ncircumsphere.htm.

I want to utilize the above criterion for being co-spheric in a graph-theoretic algorithm and want to be sure that no "exotic" exception is lurking somewhere.

Edit:

the not so "exotic" counter example of @fedja