Theorem. The intersection curve of a sphere and a plane is a circle.

Proof. We prove the theorem without the equation of the sphere. Let be the intersection curve, the radius of the sphere and be the distance of the centre of the sphere and the plane. If is an arbitrary point of , then is a right triangle. By the Pythagorean theorem,

     constant

Thus any point of the curve is in the plane at a constant distance from the point , whence is a circle.

Remark. There are two special cases of the intersection of a sphere and a plane: the empty set of points () and a single point (); these of course are not curves. In the former case one usually says that the sphere does not intersect the plane, in the latter one sometimes calls the common point a zero circle (it can be thought a circle with radius 0).