The hyperbolic lines were defined as the intersections of D with i-lines orthogonal to C. It is reasonable to ask whether intersections with other i-lines are also significant in hyperbolic geometry. Already we know that i-lines which are circles lying entirely within D are precisely the hyperbolic circles. We have also met i-lines 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 i-lines 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 and H'. The last is simply a euclidean segment. In euclidean and hyperbolic geometry, given a point P not on a line L, we can define the locus C(P,L) = {Q : P,Q on the same side of L and equidistant from L}. Of course, in euclidean geometry, this is just the line through P parallel to L.

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. proof definition Suppose that X,Y lie on the boundary of the disk, and that L is a hypercircle through X and Y. Then the width of L is the hyperbolic distance from any point of L to the hyperbolic line XY. Although we shall not attempt to identify the symmetry groups of horocycles and hypercircles, there are some useful hyperbolic symmetries. symmetry lemma Suppose that K is a hyperbolic circle, horocycle or hypercircle, and that A, B lie on K. Let h denote inversion in H, the hyperbolic perpendicular bisector of AB. Then (1) H meets K, (2) h(K) = K, (3) if K is a horocycle at X, then h(X) = X, and (4) if K is a hyperbolic circle with centre C, then h(C) = C. proof Note that, in the case of a hyperbolic circle, the hyperbolic perpendicular bisector of any chord must pass through the centre, i.e. is a diameter. We make the definition If K is a non-trivial intersection of an i-line and D, then a diameter of K is a hyperbolic line in which K inverts to itself. 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, all diamters of a horocycle pass throgh X, so X behaves like a centre. the symmetries of a horocycle. Suppose that K is a horocycle at X and that hεH(2) maps K to itself. Then h(X) = X. It follows that a hyperbolic line H is a diameter of K if and only if XεH. The first part is easy - K and C meet only in X, and h also maps C to itself. The second follows since inversion in a hyperbolic line H fixes X if and only if XεH. Our first application is to revisit the topic of hyperbolic circumcircles

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