Proof
Suppose that C does not lie on the hsegment AB.
If d(A,B) < d(A,C) or d(A,B) < d(C,B) then the
result clearly holds. After Lemma 1, this is the case
if C lies on the hline AB, but not between A and B,
so we now assume that C is not on AB
Let Ka be the hcircle with hcentre A, through C, and
let Kb be the hcircle with hcentre B, through C.
As d(A,B) ≥ d(A,C), Ka meets AB at A' between A and B.
Since A' is on Ka, d(A,A') = d(A,C).
Likewise, Kb meets AB at B' between A and B,
and, as B' is on Kb, d(B',B) = d(C,B).
Note that Ka and Kb meet at C. Since C is not on AB,
they meet again on the other side of AB. Thus A' and B'
are as shown in the (static) figure on the right.
Applying the above results, and lemma 1,
d(A,B) = d(A,B') + d(B',B)
= d(A,B') + d(C,B)
= (d(A,A')  d(A',B')) + d(C,B)
= (d(A,C)  d(A',B')) + d(C,B)
= d(A,C)  d(A',B') + d(C,B)
< d(A,C) + d(C,B)

