At https://www.cut-the-knot.org/proofs/tessellation.shtml#solution it is said, "A sphere, like a plane, can be partitioned into a union of circles plus two points. Indeed, the two points can be chosen arbitrarily. They need not be the poles, say." Can you tell me how to do this?
I tried reducing it to the problem in the plane by stereographically projecting the two points onto the plane, constructing a coaxal family of nonintersecting circles that omit these two points, and then pulling the circles onto the sphere with the inverse projection, which takes circles to circles. But this doesn't cover the "north pole" on the sphere, so three points are omitted.
Can this method be patched up somehow to work, or is a different approach needed? It the latter, can you explain it, please?