The Universality of a Diagram in Late Ancient Geometry

I'm interested in something you mentioned there which is that when you're trying to prove something you draw a diagram. How do you get the universality bit? I mean why couldn't someone just say well right but you didn't show it for a cute triangle so now draw me in a cute triangle and do the same or do a similar demonstration with that. Well in fact sometimes we do find cases like that. Not in Euclid so much but in later geometers. What you describe which is a kind of analysis of sub-cases is one of the features of late ancient geometry.