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Recall that E(A,B,C) = (s2(AC)+s2(BC)-s2(AB))/2s(AC)s(BC).
theorem
The A,B,C lie on a hyperbolic circle if and only if
-1 < E(A,B,C) < 1.
proof
The condition is equivalent to
(s2(AC)+s2(BC)-s2(AB)) < 2s(AC)s(BC), and
(s2(AC)+s2(BC)-s2(AB)) > -2s(AC)s(BC).
And hence to
(s(AC)-s(BC))2 < s2(AB), and
(s(AC)+s(BC))2 > s2(AB).
All variables are positive so we get
s(AC)-s(BC) < s(AB) and s(BC)-s(AC) < s(AB), and
s(AC)+s(BC) > s(AB).
These are clearly equivalent to
"there is a euclidean (s(AB),s(BC),s(CA)) euclidean triangle".
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