the hypercircle theorem
Suppose that L is a hyperbolic line, with boundary points X and Y,
and that P is a point which does not lie on L. Then the locus C(P,L)
is the hypercircle through X,Y and P. Also,
each hypercircle
arises in this way for some P and L.
The applet shows a green euclidean arc (a hypercircle).
If you drag P along the arc, the hyperbolic distance from
P to XY is dP,P'), and does not change.
If you drag Z within the circle, then d(Z,Z') varies, and is
equal to d(P,P') for Z in the right halfdisk only when Z is
on the arc.
proof
We may apply the Origin Lemma and a rotation so that L is
the vertical diameter, with P to the right of L. Note that
hyperbolic transformations map ilines to ilines and hence
map hypercircles to hypercircles.
Let Z have complex coordinate z. Then the hyperbolic
distance p = d(O,Z) satisfies z = tanh(½p).
Let Z' be the foot of the hyperbolic perpendicular from Z to L,
and let θ be the angle ZOZ' (in either geometry).
By the hyperbolic sine formula sin(θ) = sinh(r)/sinh(p),
where r = d(Z,Z'). By a standard identity,
sinh(p) = 2tanh(½p)/(1tanh^{2}(½p)) = 2z/(1z^{2}).
By euclidean trigonometry, sin(θ) = (z+z*)/2z since Z
is to the right of the yaxis.
Thus (z+z*) = 2sinh(r)(1z^{2}).
If we keep r constant, equal to d(P,P'), then this is an arc
of the euclidean circle through X(i), Y(i) and P, i.e. is a
hypercircle.
The last part is now obvious.

