I'm not sure if this is even possible but here it goes:
Let's assume we have a 3D Circle with 2 points on it A (1, 5, 2) B (5, 2, 1), and the circle's center O (0, 0, 0)
Is it possible to find C based on the conditions that the distance from C to the center is the radius and distances from C to A and B are equal?
On
You know that point $C$ is between $A$ and $B$, in the $AOB$ plane. Using $C=(x,y,z)$ you know that $$\vec{AO}\times\vec{CO}=\vec{CO}\times\vec{BO}$$ This accounts both for the fact that $C$ is in the $AOB$ plane, and also that the angle is the same. It yields a system with three unknown and three equations, but only two are independent. You can get the last one by using $|AO|=|CO|$, or $$x^2+y^2+z^2=1^2+5^2+2^2$$
The simplest way in my view would be to take the midpoint of $AB$ and scale it to a point on the sphere (ie, a point whose distance from $O$ is the radius).