 The proof can be extended to show that the points on a circle are all a common angular distance from one of its poles.To show that a non-trivial intersection of two spheres is a circle, assume (without loss of generality) that one sphere (with radius formula_1) is centered at the origin.Points on this sphere satisfy
Also without loss of generality, assume that the second sphere, with radius formula_3, is centered at a point on the positive x-axis, at distance formula_4 from the origin.