Skeeter and Soroban are using the fact that the distance from the center of the sphere to each of the given points is the same. Here's another way to argue that gives essentially the same equations: the general equation of a sphere is

.

To have (12,0,0) on the sphere, we must have

.

To have (0,6,0) on the sphere, we must have

.

To have (0,0,4) on the sphere, we must have

.

To have (0,0,0) on the sphere, we must have

.

That gives four equations to solve for the four values, a, b, c, and r.

When you multiply out the first three equations, each will have

plus linear terms equal to

. If you replace

by

from the fourth equation, each equation immediately reduces to a simple equation.