Show that there exist one point equidistant from four non complainer distinct points (maybe it can be solved using axis of a circle )
1. Four non-complanar points are located on a sphere. Thus you are looking for it's center.
2. If the center of the sphere is $\displaystyle C(c_1, c_2, c_3)$ then the equation of the sphere is:
$\displaystyle (x-c_1)^2+(y-c_2)^2+(z-c_3)^2=r^2$
Plug in the coordinates of the four points. You'll get a system of 4 equations which can be easily reduced to a system of 3 linear equations. Solve for $\displaystyle (c_1, c_2, c_3)$.