If L and M are the hyperbolic circles K(C,r) and K(D,s), and
d(C,D) = t, then the hyperbolic radical axis hR(L ,M) is
(1) a hyperbolic line perpendicular to CD if e-t < cosh(r)/cosh(s) < et, and
(2) the empty set otherwise.
A key step requires a lemma whose proof consists of a series of calculations.
Proof of Theorem
The proof consists of three steps:
(a) If Q ε hR(L,M), and Q* is the foot of the hyperbolic perpendicular
from Q to the hyperbolic line CD, then R ε hR(L,M).
(b) There is at most one point R in hR(L,M) which lies on CD.
(c) If the point R as in (b) exists, then the perpendicular to CD through
R is the hR(L,M).
(a) Suppose that Q* is the foot
of the hyperbolic perpendicular from a point Q
(b) Now, R ε hR(L,M) lies on CD, if and only if hp(R,L) = hp(R,M).
(c) If we have a point R as in (b), then reversing the argument in (a),