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I am looking for a rigorous geometry book where it is proved that the endpoints of a line segment are uniquely determined, the endpoint of a ray is uniquely determined, the radius and center of a circle are uniquely determined, etc. All the geometry books I have seen do not prove such results of this nature. Is there a geometry book that does prove such things?

user107952
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    Usually the endpoints of a line segment are part of the defining data of a line segment. In such a setting, it almost doesn't make sense to ask why the endpoints are uniquely determined. It's kind of like asking someone to prove that the conclusion of a formal proof is uniquely determined: the conclusion is just a name we give to one of the data used to construct a proof (namely, the last line of the proof), and therefore is uniquely determined. – diracdeltafunk Sep 06 '20 at 00:04
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    Perhaps you have a different definition of line segment, e.g. something like "a subset of $\mathbb{R}^n$ which is the image of $[0,1]$ under some affine linear map $\mathbb{R} \to \mathbb{R}^n$". In this case, you should think carefully about your definitions of "line segment", "end point", etc. and try to prove the statements you care about yourself! These things are usually not so hard to show, and would be better learned as an exercise than by reading a book. – diracdeltafunk Sep 06 '20 at 00:08
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    The construction of the center of a circle, given two points on it , is a standard exercise (and it appears in Euclid) – lulu Sep 06 '20 at 00:16
  • Well, in case you define a line segment between two points $p$ and $q$ in a normed $\mathbb R$-vectorspace as the shortest path between $p$ and $q$ then is has not to be unique (unless the norm ist strict). – Michael Hoppe Sep 06 '20 at 15:16

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