Is this a valid definition of Euclidean geometry?

Question

In trying to understand what actually constitutes a "geometry" I came across many definitions of Euclidean spaces and geometries. Euclidean space is defined as an affine space with an inner product space acting on it. I was wondering if it could be defined in an equivalent and more natural manner, by relying only on the fundamental notion of distance, without the need for an inner product.

A set $E$ is an Euclidean space iff there is a function $d:E\times E\to \mathbb R$ that satisfies the following axioms:

(1) $d(a,b)+d(c,b)\ge d(a,c)$, for every $a,b,c\in E$.

(2) $d(a,b)=d(b,a)$, for every $a,b\in E$.

(2') $d(a,b)=0$ iff $a=b$.

(3) For every $p_1,p_2$ in $E$ there always exists a unique set $P$ of points that contains $p_1,p_2$ such that for any points $a,b,c\in P$ if $d(b,c) <= d(a, c)= >d(a,b) $ then $d(a,c)=d(a,b)+d(b,c)$.

(4) For any such set $P$ and for any point $p\notin P$ there is always a unique set $P_2$ (for which (3) holds) and that contains $p$, such that for every pair $(p_1,p_2)$ where $p_1\in P$ and $p_2\in P_2$, $D \le d(p_1, p_2)$, and for every $p_1 \in P$ there exists $p_2 \in P_2$ such that $d(p_1, p_2) =D$, $D \in \mathbb{R}$.

With the variation of this last property geometry should become non-Euclidean.

First two axioms define a usual metric, third defines geodesics, and last defines parallel geodesics.

For continuity there could be a requirement that for every geodesic $P$, for any real number $r$, there always exists a pair of points $p_1, p_2$ on $P$ such that $d(p_1, p_2) =r$.

Edit : I appreciate all the comments and suggestions, I would just like to say that I wouldn't refer to this as "my axioms" foe geometry as it was not my intention to just randomly come up with some generic new axioms.

It seemed reasonable to me to ask whether various geometries could be interpreted as sets on which there is a quantitative distance relation between points that can define geodesics through triangle equality. In this case Euclidean geometry.

5. for any geodesic P and p in P and any r in R, there are p1, p2 such that d(p1, p) =d(p2, p) =r

After i posted the question i wanted to add this right away but then I didn't want to change the OG question too much.

Answer 1

Even with the most charitable interpretation of the posed question (which keeps evolving), the answer is negative. Examples are given by $\ell_p$-planes, $p\in (2,\infty)$. (I borrowed the example from this answer.)

The only thing which is not immediate is that geodesics in $\ell_p$-spaces are affine lines. The proof is not difficult, see Proposition I.1.6 in

Bridson, Martin R.; Haefliger, André, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften. 319. Berlin: Springer. xxi, 643 p. (1999). ZBL0988.53001.

where it is proven that if $B$ is a strictly convex Banach space equipped with the metric $d(x,y)=||x-y||$ then affine lines in $B$ are the only geodesics in $(B,d)$. It is also a pleasant exercise to show that an $\ell_p$-plane is not isometric to the Euclidean plane unless $p=2$.

An axiomatic system for planar Euclidean geometry based on the notion of a metric space was given by Birkhoff, see here for axioms and references.

My favorite reference is

Moise, Edwin E., Elementary geometry from an advanced standpoint., Reading, MA: Addison-Wesley. 450 p. (1990). ZBL0797.51002.

A nice and freely available treatment of Euclidean geometry from the metric viewpoint is given in

A. Petrunin, "Euclidean plane and its relatives. A minimalist introduction." Arxiv, 1302.1630.

Postulates of angle measure and similarity are missing in the set of axioms proposed by OP.

Incidentally, the following is a cute open problem due to Keith Burns:

Suppose that $X$ is a Riemannian surface (complete, simply connected, without conjugate points) which satisfies the Playfair's axiom. Does it follow that $X$ is flat?

In this setting, all Birkhoff's postulates hold except for, possibly, the similarity postulate.

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Edit. Here is a clean interpretation of OP's question.

Let $(X,d)$ be a metric space. (No need to define this, since such a definition is a standard undergraduate material.)

Definition. 1. A map $\gamma: {\mathbb R}\to (X,d)$ is called an isometric embedding if $d(\gamma(s), \gamma(t))=|s-t|$ for all $s, t\in {\mathbb R}$.

2. A line in $(X,d)$ is the image of an isometric embedding $\gamma: {\mathbb R}\to (X,d)$.

Now, one can state OP's axioms:

A1. $(X,d)$ is a metric space containing at least two distinct lines. (This axiom was presumably simply forgotten by OP.)

A2. Every two distinct points in $(X,d)$ belong to one and only one line.

A3. For every line $L$ in $(X,d)$ and a point $p\in X\setminus L$ there is one and only one line $M$ in $(X,d)$, containing $p$ and disjoint from $L$.