TITLE: Bolyai's construction
QUESTION [6 upvotes]: Here is Bolyai's construction (in Klein model), which I learned recently from answer of Will Jagy to this question.
It is a Compass-and-straightedge construction of asymptotically parallel line in absolute geometry.

Do you know an elementary proof showing that Bolyai's construction really does what it suppose to? 
"Elementary" means without calculations and without referring to the models.
Comments: 

I know a simple proof in Klein model (thanks to A. Akopyan). (It is easy to guess from the picture above. Draw one more vertical line and note that cross ratios for two quadruples of points on two asymptotically parallel lines is the same.)
If there is a simple argument, I would include it in the course on Foundation of geometry which I'm teaching at the moment.
At the moment I do not know a simple argument in the Poincaré disk model.

REPLY [4 votes]: There is a proof in Introduction To Non Euclidean Geometry by Harold E. Wolfe
It is an elementary proof without calculations or referring a models.
You can get a copy here:
http://www.archive.org/details/introductiontono031680mbp