the hypercircle theorem

 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 half-disk 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 i-lines to i-lines 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)/(1-tanh2(½p)) = 2|z|/(1-|z|2). By euclidean trigonometry, sin(θ) = (z+z*)/2|z| since Z is to the right of the y-axis. Thus (z+z*) = 2sinh(r)(1-|z|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.