Theorem HC1 Suppose that K is a hyperbolic circle with centre C and radius r, and that the chord AB of K passes through the fixed point P. Let d(P,A) = a, d(P,B) = b and d(P,C) = c. Then sinh(a+b)/(sinh(a)+sinh(b)) = cosh(r)/cosh(c) if P is inside K, sinh(a+b)/(sinh(a)+sinh(b)) = 1 if P is on K, sinh(a+b)/(sinh(a)+sinh(b)) = cosh(c)/cosh(r) if P is outside K.


Proof
Suppose first that P is inside K.
If P is on K, then P = A or B. We may as well assume P = A.
The proof for P outside K is very similar to the first part,

