a horocycle theorem
Suppose that A(0),..,A(m) lie in order on a horocycle, and
let l(i) = d(A(i-1),A(i)), i = 1,.,m, L = d(A(0),A(m)). Then
s(L) = s(l(1))+..+s(l(m)).
proof
Suppose that A,B,C lie on a horocycle, with C on the finite arc AB.
For a horocycle, J(A,B,C) = 0, so that E2(A,B,C) = 1.
As C is on the finite arc, the theorem gives E(A,B,C) = -1.
But E(A,B,C) = (s2(AC)+s2(BC)-s2(AB))/2(s(AC)s(BC).
It follows that s(AB) = s(AC)+s(BC).
The general result follows by applying this in turn to
A(0),A(1),A(2), to A(0)A(2),A(3),.., to A(0)A(m-1),A(m).
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