My proof reasons like this: given a point in 3 space, call it p_1, there is all of 3 space except for p_1 where we can choose for the center of a sphere to be which goes through p_1 (and in fact choosing a center, c, and requiring the sphere to go through p_1 will uniquely determine a sphere). Now, choose a second point p_2, and notice this: if the sphere is to go through both p_1 and p_2 then the center of the sphere must be the same distance from p_1 and p_2. What is the shape describing all points in 3 space which are the same distance from p_1 as they are from p_2?