Let's remove one axiom from Euclidean geometry

[Vicol]

I'm wondering what could happen if we remove one axiom from Euclidean geometry. What are the conseqences? For example - how would space without postulate "To describe a cicle with any centre and distance" look like?

 

[jedishrfu]

The way you'd have to approach this is how specific theorems of Euclidean Geometry are affected when you remove a postulate.

http://mathworld.wolfram.com/EuclidsPostulates.html

As am example, there was a time when people tried to remove the parallel postulate and prove it from the other four but they all failed. Later it was determined that new geometries resulted from a relaxation of the parallel postulate.

https://en.wikipedia.org/wiki/Parallel_postulate

Off hand, I can't see the affects that removing the circle or allowing for multiple circles would have on the geometry. Perhaps looking at ordered geometry or absolute geometry will give you some insight:

https://en.wikipedia.org/wiki/Ordered_geometry

https://en.wikipedia.org/wiki/Absolute_geometry