Suppose that A and B lie on a hyperbolic circle K.
The hyperbolic line AB must contain points outside K,
so the hyperbolic segment AB cannot meet K between
A and B.
It follows that the line AB divides the circle K into two
If we suppose, further, that AB does not pass through
In euclidean geometry, we are familiar with results on
Since we are largely concerned with hyperbolic length
the hybrid angle theorem
Suppose that A,B,C are points in the disk, with A ≠ B.
If A,B,C lie on a circle K with centre O, then
cos(e(ACB)) = (s2(AC)+s2(BC)-s2(AB))/2s(AC)s(BC).
Remark Since hyperbolic length of XY, and hence s(XY) are
The function E gives yet another characterization of circumcircles.
More importantly, it aso allows us to describe the major and minor
characterisation of minor and major arcs
We will also require the following algebraic result :
the F-H relation
The proof is simply an algebraic manipulation from the given formulae
|Note Since the (hyperbolic) centre
is O, K is a also euclidean circle with centre O.
We shall begin by looking at the case where AB is a diameter.
the hyperbolic semi-circle theorem
We can, of course also state the result in hybrid form:
the hybrid semi-circle theorem
|We may restate the condition for C on K as
if and only if the euclidean triangle
|The remaining theorems on points on hyperbolic arcs follow quickly|