However, I still made my own program for this. It supports all the above stated required rules, but none of the Extra challenges. Maybe I will update my program and add the extra challenges, so please dont just steal my program and then add the extra challenges by your own, to also participate here.
So, here is my code, it first asks if the hypotenusis of each segment should be displayed (extra feature), and then it calculates to infinity, after some iterations you dont see the outer lines, because it does not automatically zoom, which is a huge disadvantage. But I will add this feature later, it should not be that hard, because then you only need to track the outer hypotinusis points in a list or something.
But without all these extra features, my program is very good in terms of speed (for TI Basic)
TI Basic Spiral of Theodorus (216 Bytes)
:Disp "SHOW", "HYPOTENUSIS?", "", "(1=YES) / (0=NO)", "
If you want to have a quadratic view, you will have to use these factors for the window size: