(1)The h-centre of an h-circle lies within the circle.
(2)The h-centre and h-measure of an h-circle are unique.
Proof
(2)Suppose that the h-circle K can be describes as K(P,r) and as K(Q,s),
Then we know from part (1) that P and Q lies within K.
Then, so is the h-measure,
since the h-measure is D(P,T) for any T on K.
(1)Suppose that R is outside K, and let E be the euclidean centre of K.
Then the h-line ER contains the interior point E, so meets K twice.
Let the intersections be P and Q. We may label these so that Q lies
between R and P. But then, by Theorem 4, D(P,R) > D(Q,R), so that
R is not an h-centre for K.
with P ≠ Q.
The h-line PQ contains the interior point P, so by Basic Fact (3), meets K twice.
Let the intersections be R and S.
As K = K(P,r), D(R,P) = D(S,P), i.e. P is the h-midpoint of RS.
Similarly, Q is the h-midpoint, contradicting the assumption that P ≠ Q.
Thus the h-centre is unique.
