A circle does not touch a circle at more points than one, whether it touch it internally or externally.
Proof
Suppose two circles touch at two points A, B. By III.11
(internal) or III.12 (external), both A and B lie on the line
joining the centres. Thus this line cuts each circle in two
points, making it a diameter of each. But then AB is a chord of
each circle equal in length to the diameter — so A and B are
antipodal points on each circle, and both circles share centre and
diameter, contradicting III.5/III.6.