Originally Posted by

**Plato** I am not sure why you say it is simple or there should be two possible answers.

There is one answer if you want the three points to be on a *great circle* of the sphere. Otherwise, there is a whole line of centers.

If $\displaystyle P,~Q,~\&~R$ are the three points then the points of the plane $\displaystyle \pi_1$ that is the perpendicular bisector of $\displaystyle \overline{PQ}$ are equidistant from $\displaystyle P~\&~Q$.

Same said for $\displaystyle \pi_2$ with respect to $\displaystyle P~\&~R$.

The points on the line $\displaystyle \pi_1\cap\pi_2$ are equidistant from $\displaystyle P,~Q,~\&~R$.

That is a messy problem to solve algebraically.