I took a lot of Math courses, including a geometry grad course. So I’ve known all about Elliptic and Hyperbolic Geometry.

But recently I came across the Wikipedia article on Pasch’s axiom (1882). An “obvious” property of plane Geometry that Euclid even used, despite it not being stated as an axiom or proven from his axioms. (It can’t.)

In short: You run a straight line thru a triangle not hitting any vertex. If you cross one side you cross one other side. See the link for more info.

Pasch saw the problem and proved it’s independence of Euclidean Geometry.

I never heard about this. Later formalizations of Plane Geometry included it. It seems to be the same as something called the plane separation axiom.

Apparently “order” is the key, something which classic Geometry doesn’t pay much attention to.

What else did I miss that is of that flavor: so clearly obviously true but Euclid and others didn’t see it had to be stated?

Googling reveals surprisingly few pages on this. So clearly not as well known as the Big non-Euclidean Geometries.