Throughout this page, we will consider stereographic projection p with vertex N from the sphere S to the plane P. Lemma
If the circle C on S passes through N, then the projection p(C)
Proof
We know from Theorem 1 that L = p(C) is a line.
The picture is symmetric about NP, the diameter of C through N,


Theorem 2 Stereographic projection preserves angles.


Proof Since the angle between curves on P is defined in terms of tangents, it is enough to consider the case of two lines L and M on P, meeting at Q with angle a. Suppose that these correspond to the circles C and D on S. They meet at P = p^{1}(Q). Since L and M are lines, C and D also meet in N.
By the Lemma, the tangents to C and D at N are parallel to L and M.
By an earlier remark, the angle beween two circles at both 
Stereographic projection 