I see, yes, Euclid's axioms are not the most intuitive approach to different geometries. What is nice is to have the tools to examine what the properties of a

I see, yes, Euclid's axioms are not the most intuitive approach to different geometries. What is nice is to have the tools to examine what the properties of a given geometry are, and given that geometry is a matter of curvature, it's not going to be decided by the tangent plane, it's going to be decided by the second derivative. You can get at that explicitly by embedding your space in a flat space like R^N and looking at the second derivative, or you can do intrinsic operations like parallel transport. E.g. look at small variations in the tangent plane from point to point. But the second derivative is key. Geometric algebra lives in the cotangent plane so it alone is not going to detect issues of curvature in your underlying space. This is true even though a lot of important calculations about differentials and volume elements are happening in that cotangent plane, so it's an important thing to get right, but it can't detect issues of curvature and thus it can't 'prove' the pythagorean theorem, which is a flatness statement.