The hyperbolic lines were defined as the intersections of D with ilines orthogonal to C. It is reasonable to ask whether intersections with other ilines are also significant in hyperbolic geometry. Already we know that ilines which are circles lying entirely within D are precisely the hyperbolic circles. We have also met ilines which are circles touching C internally at a point X. These are the horocycles at X. They arose as the boundaries of regions containing points C for which ABC is a hyperbolic cyclic triangle. In the course of our work on Ptolemy's Theorem, we met the remaining family. These are the intersections with ilines which cut C twice, but are not orthogonal to C. These are the hypercircles. In the context of Ptolemy's Theorem, they simply arose as part of the class of functions to which the theorem applied. They appear in a much more interesting way.
The figure on the right shows a hyperbolic circle K, a horocycle E and hypercircles H
In euclidean and hyperbolic geometry, given a point P not on a line L, we can define


the hypercircle theorem Suppose that L is a hyperbolic line, with boundary points X and Y, and that P is a point which does not lie on L. Then the locus C(P,L) is the hypercircle through X,Y and P. Also, each hypercircle arises in this way for some P and L.
definition
Although we shall not attempt to identify the symmetry groups of horocycles and
symmetry lemma
Note that, in the case of a hyperbolic circle, the hyperbolic perpendicular bisector
definition The lemma shows that the bisector of any chord will be a diameter in this sense
For a horocycle at X, the diameters obtained in this way all pass through X. In fact,
the symmetries of a horocycle.
The first part is easy  K and C meet only in X, and h also maps C to itself. Our first application is to revisit the topic of hyperbolic circumcircles 