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?
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.
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:
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