# Properties of hyperbolic circles

 We have defined the h-circle K(γ,r) as the locus {z : D(z,γ) = r}, and have seen that an h-circle is a circle lying within D. However, it is not clear that the h-centre and h-measure of an h-circle are well-defined, i.e. that, if K(γ,r) = K(δ,s), then γ = δ and r = s. It is not even clear that the h-centre 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 h-circles, 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 h-centre and any point on K(γ,r). To proceed, we need some results from euclidean geometry: Basic facts about euclidean circles any three non-collinear 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 h-circle is a circle, and an h-line is either a segment of a line, or an arc of a circle, we have the Basic facts about h-circles an h-line and an h-circle meet in at most two points, two distinct h-circles meet in at most two points, if an h-line contains an interior point P of an h-circle, then the h-line and h-circle 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 h-circle K lies within D, so that any h-line 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 h-circle in hyperbolic terms: a point P lies in the interior of an h-circle K if and only if an h-line 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 h-circle 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 h-circle K(P,r) is symmetric about the h-line H if and only if P lies on H. Proof Let h denote h-inversion 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 h-centre 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 h-line and h-circle meet once. As in euclidean geometry, we make the Definition If the h-line H meets the h-circle K in a single point P, then H is a hyperbolic tangent (h-tangent) to K at P. The Hyperbolic Tangents Theorem If P lies on the h-circle K = K(Q,r), then (1) there is a unique h-tangent to K at P, and (2) the h-tangent is perpendicular to the h-line PQ. Proof By the Origin Lemma, there is an h-inversion h mapping P to 0. This maps K to the h-circle K(h(Q),r) passing through 0. Since any h-line through 0 is a diameter, there is a unique h-line tangent to K(h(Q),r). Applying h-1, we get a unique h-tangent J at P. Let hQ denote h-inversion in the h-line PQ. Then hQ(K) = K and hQ(P) = P. SInce the h-tangent is unique, we must have hQ(J) = J. Thus J is perpendicular to PQ (see symmetry of h-lines).