Theorem II2
(1) The points A,B,C,D lie on an iline if and only if (A,B,C,D) is real,
(2) The points C,D lie on the same isegment AB if and only if (A,B,C,D) > 0.
Proof
The key steps are the observations that a Mobius transformation t:
maps ilines to ilines,
maps isegments to isegments,
preserves inversive crossratio, and
can be chosen to map A,B,D to 0,∞,1.
Also, the only iline through 0,∞,1 is R, the extended real line.
Then, t maps the iline through A,B,D to the extended real line.
Suppose that t maps C to the point with complex coordinate c.
Then (A,B,C,D) = t(A,B,C,D) = (0,∞,c,1) = (c0)/(10) = c, so that
C is on the iline through A,B,D if and only if c is on R, i.e. c is real since C
and D are distinct, so c ≠ ∞.
For (2), note that 1 lies on the isegment (0,∞), and c lies on this
if and only if c > 0. Applying t^{1}, we see that C,D lie on the same
isegment AB if and only if c = (A,B,C,D) > 0.

