further theorems on hyperbolic circles

We observe that, by invoking the cosine rule in the euclidean case,
we can get analogous results there. Indeed, all we have to do is to
replace E by E*(A,B,C) = (x²+y²-z²)/2xy, where x = |AC|, y = |BC|,
z = |AB|.

One important difference is that, in theorem 2, we need to assume
that at least one of ΔABC and ΔABD has a hyperbolic circumcircle.
This is automatic in the euclidean case.

hyperbolic circle theorem 1 Suppose that A and B lie on a hyperbolic circle K. Then the segment AB divides K into two arcs. Let C and D be points of K distinct from A and B. Suppose that AB is not a diameter of K. (1) C,D lie on the same arc if and only if E(A,B,C)=E(A,B,D). (2) C,D lie on opposite arcs if and only if E(A,B,C)=-E(A,B,D). proof 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 Naturally, there are corresponding hybrid theorems. These may be summarized as the hybrid arcs theorems Provided that at least one of the hyperbolic triangles involved has a circumcircle, the arc properties of a hyperbolic quadrilateral with sides of hyperbolic length (r,s,t,u) correspond to those of the euclidean quadrilateral with sides of length (s(r),s(s),s(t),s(u)) These follow as E for the hyperbolic quadrilateral gives the cosine of an angle for the euclidean quadrilateral. Remark We used E in the statements since that suggested the line of proof. This leads to some results on cyclic quadrilaterals.

hyperbolic geometry