Is there a compass and straightedge construction of parallel lines in hyperbolic geometry? That is, given a line and a point not on the line, construct a line parallel to the given line.

The quickest way to get you started is to refer you to my article, reference [5] (a pdf) on http://en.wikipedia.org/wiki/Squaring_the_circle and then to the fourth edition (2008) of Marvin Jay Greenberg's book, which is reference [6]. I'm guessing what you want is Bolyai's construction, given a line and a point off the line, of the two rays through the point that are asymptotic to the line, one in each direction. When I wrote the article, I relied on an earlier edition of Marvin's book, along with The Foundations of Geometry and the NonEuclidean Plane by George E. Martin, which has a nice little section at the very end. There is also, now, Geometry: Euclid and Beyond by Robin Hartshorne. The most complete reference I know on constructions is in Russian, by Smogorshevski, other very helpful books by Kagan and by Nestorovich. Of course, at this point I have my own versions of it all. 


So you mean you want to see a compass and straight edge construction in Euclidean geometry of a circle passing through 2 given points and perpendicular to the given circle (which contains one of the points)? I believe that this construction is given in the geometry book by Robin Hartshorne, excellent book by the way. 

