proof
The equivalence of the conditions is clear by result 1.
Now suppose that C,D lie on a hyperbolic circle K through A,B.
Case 1 : C,D lie on opposite arcs of K.
Then C*,D* lie on opposite arcs of the image circle by result 1
Then ACBD is a convex cyclic hyperbolic quadrilateral, and
A*C*B*D* is a convex cyclic euclidean quadrilateral.
From the definitions of C*,D*, we have
|A*C*|=s(AC), |A*D*|=s(AD), |B*C*|=s(BD), |B*D*|=s(BD).
We also have |A*B*|=s(AB).
If we apply the euclidean and hyperbolic verions of ptolemy
to A*C*B*D* and to ABCD, respectively, we see that five of
the six quantities agree, and so |C*D*|=s(CD), as required.
Case 2 : C,D lie on the same arc of K.
We may clearly choose the labels so that the points A,B,C,D
lie in this order on K, so ABCD is convex cyclic.
By result 2, A*,B*,C*,D* lie in this order on the image so
A*C*D*B* is convex.
Then ptolemy's theorems give result
as in Case 1.