Theorem II2
(1) The points A,B,C,D lie on an i-line if and only if (A,B,C,D) is real,
(2) The points C,D lie on the same i-segment AB if and only if (A,B,C,D) > 0.
Proof
The key steps are the observations that a Mobius transformation t:
maps i-lines to i-lines,
maps i-segments to i-segments,
preserves inversive cross-ratio, and
can be chosen to map A,B,D to 0,∞,1.
Also, the only i-line through 0,∞,1 is R, the extended real line.
Then, t maps the i-line 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) = (c-0)/(1-0) = c, so that
C is on the i-line 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 i-segment (0,∞), and c lies on this
if and only if c > 0. Applying t-1, we see that C,D lie on the same
i-segment AB if and only if c = (A,B,C,D) > 0.
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