Theorem
If a plane F cuts a sphere S nontrivially,


Proof Suppose that the point P lies on C, the curve of intersection. Let Q be the foot of the perpendicular to F from O, the centre of S. Then DOPQ is rightangled at Q. Hence OP^{2} = OQ^{2} + QP^{2}, so that QP^{2} = OP^{2}  OQ^{2}. But OP is the radius of S, and OQ is the distance from O to F, so are constant. Thus QP is constant, so C is a circle. 
Stereographic projection 