Many results about hyperbolic lines have analogues for complete hyperbolic lines.
For example, we know that, if hyperbolic lines K, L meet at X in D, then there are
exactly two hyperbolic lines which bisect the angles at X. If H is such a line, then
it is easy to see that inversion in H interchanges K and L. It is also easy to see
that an H with the latter property must bisect a pair of angles at X.
definition
If hyperbolic lines H,K,L are such that inversion in H swaps K and L, then we say
that H is an axis (of symmetry) for the pair K,L.
Theorem HA4
Suppose that K,L are complete hyperbolic lines.
If K,L meet in D then K,L have two axes, otherwise they have one.
proof


We could have chosen to characterize the axes of K,L as the locus of points equidistant
from both. Although this would have simplified some calculations, it would not have been
at all clear that the locus would always consist of one or two hyperbolic lines.
To show that the ideas are related we prove
Theorem HA5
The point PεD is equidistant from the hyperbolic lines K and L if and only if
P lies on an axis for K,L.
proof
This leads to results about hyperbolic circles and horocycles which touch two or more
hyperbolic lines. Suppose that P is a point in D, and that K is a hyperbolic line which
does not pass through P. Let U be the foot of the perpendicular from P to K. Then the
hyperbolic circle through U with centre P touches K since PU is perpendicular to K.
It follows that there is a hyperbolic circle with centre P and touching the hyperbolic lines
K, L if and only if P lies on an axis for K,L, by theorem HA5. Thus, we have
Corollary HA5.1
Suppose that K,L are hyperbolic lines, and that PεD.
There is a hyperbolic circle with centre P touching K,L if and only if
P lies on an axis for K,L.
To allow us to extend this to horocycles, we need more information about touching
horocyles and hyperbolic lines.
tangency lemma
Suppose that X is a point on C, and that K is a hyperbolic line not through X.
Then there is a unique horocycle J at X touching K.
Indeed, J touches K at the foot of the perpendicular from X to K.
We cannot measure the hyperbolic length of a segment AX with X on C, so we cannot
have an analogoue of HA5. We do have an analogue of HA5.1.
Theorem HA6
Suppose that K,L are hyperbolic lines, and that XεC.
There is a horocycle at X touching K,L if and only if
X lies on an axis for K,L.
proof
This allows us to generalise the results about incircles and excircles as follows:
Theorem HA7
Suppose that K,L,M are nonconcurrent hyperbolic lines, that
H is an axis for K,L, H' an axis for L,M and that
H,H' meet at PεE, then
(1) P lies on an axis for K,M,
(2) if PεD, then there is a hyperbolic circle centre P touching K,L,M,
(3) if PεC, then there is a horocycle at P touching K,L,M.
proof
Using HA5.1 if PεD, or HA6 if PεC, we see that there is a hyperbolic circle centre P,
or a horocycle at P touching K,L. It touches L at the foot of the perpendicular from P.
It follows that it also touches M (applying the results to L,M). Thus we have a circle
or horocycle touching K,M, and the results follow.
Of course, the existence depends on at least two of the axes meeting. In some cases,
this can be guaranteed. For example, if K,L,M are the extended sides of a hypebolic
triangle, then the internal angle bisectors each cross the interior of the triangle, so
any two meet. We then get the incircle  a hyperbolic circle as it cannot reach the disk
boundary.This works even when we allow asymptotic triangles.

