I want to show that there are circles in the hyperbolic plane, which cannot be inscribed in a triangle. Is it enough for me to say that, since the largest possible triangle is an ideal triangle and the largest radius for a circle inscribed in an ideal triangle is ${1\over 2}ln(3)$, then any circle with a greater radius cannot be inscribed in a triangle? Am I missing something or is this argumentation correct?
2026-02-22 19:50:55.1771789855
Is this proof that there are circles in hyperbolic plane that cannot be inscribed in a triangle correct?
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Yes, the argument is sound.
Of course strictly speaking it depends on what you can build upon, to make sure the argument is not cyclic. But all of the things you need to assume appear readily provable using more fundamental arguments, so while I'm sure this would require more work in a formal proof system, I as a human am content with the argument.
Essentially you need to show that the incircle is the largest possible circle in any triangle, and that any triangle which is fully contained inside another triangle cannot have a larger incircle than the containing one, and that every other triangle can be contained inside an ideal triangle. I think you see how these facts need to be connected to show your claim in more detail, and also should have an idea as to why each of them is true. If you are certain you could fill in the gaps, that's fine. If you are less certain, I encourage you to actually do so.