We have defined the hcircle K(γ,r) as the locus {z : D(z,γ) = r},
and have seen that an hcircle is a circle lying within D. However,
it is not clear that the hcentre and hmeasure of an hcircle are welldefined,
i.e. that, if K(γ,r) = K(δ,s), then γ = δ and r = s.
It is not even clear that the hcentre lies inside the circle!
Indeed, if we define a euclidean circle as a locus {z : z γ = r}, then
the problems arise even in euclidean geometry.
The nicest way to resolve these problems is to use the theory of apollonian families.
In the proof of the Lemma on hcircles, we observed that K(γ,r) belongs to the
apollonian family A(γ,1/γ*) if γ ≠ 0, and to A(0,∞) otherwise. Note that C
belongs to the family in either case. By the Common Inverses Theorem from
inversive geometry, two disjoint circles belong to a unique apollonian family,
so γ is uniquely defined by the locus
K(γ,r). It follows easily that the radius
is unique since it is determined by the hcentre and any point on K(γ,r).
To proceed, we need some results from euclidean geometry:
Basic facts about euclidean circles
 any three noncollinear points lie on a unique circle,
 a line and circle meet in at most two points,
 two distinct circles meet in at most two points.
Since an hcircle is a circle, and an hline is either a segment of a line,
or an arc of a circle, we have the
Basic facts about hcircles
 an hline and an hcircle meet in at most two points,
 two distinct hcircles meet in at most two points,
 if an hline contains an interior point P of an hcircle,
then the hline and hcircle meet in two distinct points,
with P between these points.
Parts (1) and (2) are immediate from euclidean facts (2) and (3).
For (3), we need to observe that an hcircle K lies within D, so that any
hline H must contain points outside K (nearer the boundary C). Thus,
if H contains an interior point P, then it meets K once on each side of P.
Fact (3) characterizes the interior of an hcircle in hyperbolic terms:
a point P lies in the interior of an hcircle K if and only if an hline
through P cuts K at points Q and R, with P between Q and R.
It follows that any hyperbolic transformation t maps the interior of an
hcircle K
to the interior of t(K).


In euclidean geometry, the centre plays a vital role in dicussion of
the symmetries of the circle, and of tangents.
Theorem 4
An hcircle K(P,r) is symmetric about the hline H
if and only if P lies on H.
Proof
Let h denote hinversion in H.
By the Hyperbolic Circle Theorem, h(K(P,r)) = K(h(P),r).
Thus, we have symmetry about H if and only if K(P,r) = K(h(P),r).
Since the hcentre is unique, this occurs if and only if
h(P) = P,
i.e. P is on H.
Basic Fact (2) leaves the possibility that an hline and hcircle meet once.
As in euclidean geometry, we make the
Definition
If the hline H meets the hcircle K in a single point P,
then H is a hyperbolic tangent (htangent) to K at P.
The Hyperbolic Tangents Theorem
If P lies on the hcircle K = K(Q,r), then
(1) there is a unique htangent to K at P, and
(2) the htangent is perpendicular to the hline PQ.
Proof
By the Origin Lemma, there is an hinversion h mapping P to 0.
This maps K to the hcircle K(h(Q),r) passing through 0.
Since any hline through 0 is a diameter, there is a unique hline
tangent to K(h(Q),r). Applying h^{1}, we get a
unique htangent J at P.
Let h_{Q} denote hinversion in the hline PQ. Then h_{Q}(K) = K
and h_{Q}(P) = P. SInce the htangent is unique, we must have h_{Q}(J) = J.
Thus J is perpendicular to PQ (see symmetry of hlines).

