# Studying Euclidean geometry using hyperbolic criteria

You’ve spent your whole life in the hyperbolic plane. It’s second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.

But recently a good friend named Euclid has raised doubts about the fifth postulate of Poincaré’s Elements. This postulate is the obvious statement that given a line $L$ and a point $p$ not on $L$ there are at least two lines through $p$ that do not meet $L$. Your friend wonders what it would be like if this assertion were replaced with the following: given a line $L$ and a point $p$ not on $L$, there is exactly one line through $p$ that does not meet $L$.

You begin investigating this Euclidean geometry, but you find it utterly impossible to visualize intrinsically. You decide your only hope is to find a model of this geometry within your familiar hyperbolic plane.

What model do you build?

I do not know if there’s a satisfying answer to this question, but maybe it’s entertaining to try to imagine. For clarity, we Euclidean creatures have built models like the upper-half plane model or the unit-disc model to visualize hyperbolic geometry within a Euclidean domain. I’m wondering what the reverse would be.