  Given two points A,B in the plane, the locus of all points P for which the
angles at A and B in the triangle A,B,P have a constant sum is a circular arc,
by Thales' theorem. We show that the difference of these angles is kept a
constant by points P on a hyperbola (albeit with foci different from A and B).
Whereas hyperbolae are well-known to maintain a constant difference between the
distances to their foci, the above angle property seems not to be widely known.
The question was motivated by recent work by Alegr\'ia et al. and De Berg et
al. on Voronoi diagrams of turning rays.
