hyperbolic circle theorem 2
Suppose that A,B,C,D are distinct points, and that
A,B,C lie on a hyperbolic circle K.
(1) If C and D lie on the same side of the line AB, and
E(A,B,C)=E(A,B,D), then D lies on the same arc as C.
(2) If C and D lie on opposite sides of the line AB, and
E(A,B,C)=-E(A,B,D), then D lies on the opposite arc from C.
proof
We assume that the picture has been transformed so that K
has hyperbolic (and hence euclidean) centre O.
If E(A,B,C) = ±E(A,B,D), then, by the E-H relation,
H(s(AB),s(BC),s(CA))/4s2(AC)s2(BC)= H((s(AB),s(BD),s(DA))/4s2(AD)s2(BD).
The circle K has hyperbolic radius r given by version 2.
This version also shows that, as K exists, the left hand side above is positive.
It follows that the right hand side is positive, so A,B,D lie on a hyperbolic circle.
Further, this also has hyperbolic radius r, and passes through a and B.
Now, a hyperbolic circle of hyperbolic radius r through A and B must have
hyperbolic centre on the hyperbolic bisector of AB and on the hyperbolic
circle K(A,r). There are thus two possibilities, K itself and K*, the result of
inverting K in the hyperbolic line AB.
We want to show that D lies on K. Suppose that D lies on K*. Inverting D in
the line AB, we get a point D* on K, and on the opposite side of AB from D.
Since inversion preserves hyperbolic length, E(A,B,D) = E(A,B,D*).
(1) here D and C lie on the same side of AB, so C,D* lie on opposite sides,
and hence on opposite arcs. Then, by Theorem 1, E(A,B,C)=-E(A,B,D*).
But E(A,B,C)=E(A,B,D), so each is zero, and AB is a diameter. But then
K = K*, so D also lies on K.
(2) is very similar.
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