For those who don’t know what it is and feel lazy to look up in wiki — stereographic projection is defined as follows. Imagine a plane in and a sphere of unit radius ‘lying’ on this plane such that the ‘south pole’ of the sphere is the origin. Let be the north pole. Then stereographic projection maps any point of the sphere to the point at which ray intersects the plane. itself is mapped to “infinity point” of the plane.
Circle-preserving property means that, under this projection, circles on the sphere become circles on the plane (straight lines are also circles with infinite radius) and vice versa.
It is a good exercise for a schooler, planimetry is the only prerequisite (triangles similarity and angles in a circle). A good way of thinking about that is a beam of projector.
All the proof is essentially the image on p. 6.