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We have the familiar euclidean result
Theorem EC1
Suppose that K is a euclidean circle with centre C and radius r,
and that the chord AB of K passes through the fixed point P.
|PA||PB| = r2 - |PC|2 if P is inside K,
|PA||PB| = 0 if P is on K, and
|PA||PB| = |PC|2 - r2 if P is outside K.
This can be proved quite easily using the Cosine Formula.
Note that this implies that the value of |PA||PB| is independent of the choice
of the chord AB as long as P lies on the line AB.
Following this, it is natural to define p(P,K), the power of P
with respect to K
as |PC|2 - r2, where r is the radius and C the centre of K.
For two circles K and L, we define the radical axis
of the circles K and L as
the locus R(K,L) = {Q : p(Q,K) = p(Q,L)}.
Theorem EC2
If K and L are circles with centres C and D, then the radical axis R(K ,L) is
(1) a line perpendicular to CD if C ≠ D, and
(2) the empty set otherwise.
This leads almost imediately to
Theorem EC3
Suppose that K, L and M are circles such that the radical axes
R(K,L), R(L,M) and R(M,K) exist. If any two of the axes
intersect, then all three are concurrent.
Of course, in euclidean geometry, lines either meet or are parallel, so we
can rephrase the result as "the radical axes are parallel or concurrent".
Here, we prove analogues for hyperbolic circles.
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
This allows us to calculate the length of the hyperbolic tangents from a
point P external to a hyperbolic circle L = K(C,r). If a hyperbolic tangent
from P touches L at A. Then this cuts L in the "chord" AA. The Theorem
shows that if a = d(P,A), and c = d(P,C), then
sinh(a+a)/(sinh(a)+sinh(a)) = cosh(c)/cosh(r), so that
cosh(a) = cosh(c)/cosh(r). This can also be obtained from the hyperbolic
version of Pythagoras's Theorem.
We now define the hyperbolic power of the point P with respect to the
hyperbolic
circle L = K(C,r) as hp(P,L) = cosh(d(P,C))/cosh(r) and the
hyperbolic radical axis of two hyperbolic circles L and M
as the locus
hR(L,M) = {Q : hp(Q,L) = hp(Q,M)}.
Theorem HC2
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.
proof
If hyperbolic circles L and M meet at A and B, then we have
hR(A,L) = 1 = hR(A,M) and hR(B,L) = 1 = hR(B,M)
Thus A and B lie in hR(L,M). From the theorem, the axis is a hyperbolic
line, so it must be the hyperbolic line AB.
If A = B, so the hyperbolic circles touch at A. Then A lies on the radical axis
and the perpendicularity condition shows that the radical axis is the common
hyperbolic tangent at A.
Theorem HC2 leads almost imediately to
Theorem HC3
Suppose that K, L and M are hyperbolic circles such that the hyperbolic radical
axes
hR(K,L), hR(L,M) and hR(M,K) exist. If any two of the axes intersect,
then all three are concurrent.
Proof
If two of the radical axes meet, we may as well assume that hR(K,L) and
hR(L,M)
meet at a point Q. Then hp(Q,K) = hp(Q,L) and hp(Q,L) = hp(Q,M). Then
hp(Q,L) = hp(Q,M), so Q also lies on hR(L,M) as required.
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