Wow, this is so close. Thank you so much. I will have to study this for awhile. I think "If[IntegerQ[#/(π/2)], 1, Sqrt[2]]" is causing some diagonals to be drawn incorrectly?? Test this set: Select[Tuples[{0, 45, 90, 135, 180, 225, 270, 315}, 5] , Total[#] == (5 + 2)*180 &] --all the possible angle sets for order-3 polytans.

@HyperGroups: I'll take a look at the code, if I can. Thanks. EDIT: hmmm, I seems to be able to draw a 90/45/44 triangle with that demo. ;)

@Silva: Ultimately, yes, but initially no. For now, I just want to draw everything, as I did in the image.

Thanks! Your last sentence isn't yet clear to me, so I'll have to play with your code.

@Artes: Tuples[] looks promising. Subsets[] is no good, as it doesn't produce repetitions -- try Subsets[{1,2},3]

I can finally add a comment! (Firefox was giving me headaches) Thanks to all for the thoughtful answers.