Thank you for giving the solution to the N.A.Court problem I posted.
After several attempts I gave in and used brute force, that is trigonometry.I implemented your construction using Geometer's Sketchpad. It is an elegant and in retrospect a "simple" solution, reminds me the problem sets found in Yaglom's Geometric Transformations.
I could not quite figure out the solution suggested using circle inversion, but I will keep trying.
Tamas Varhegyi