In KG Binmore's "Topological Ideas" he says
The geometric terms which appear in Hilbert's axioms are the words point, line, lie on, between and congruent. To show $\mathbb{R}^2$ is a model for Euclidean plane geometry one has to give a precise definition of each of these words in terms of $\mathbb{R}^2$ and then prove each of Hilbert's axioms for Euclidean plane geometry as a theorem in $\mathbb{R}^2$... Interested readers ill find the book Elementary geometry from an advanced standpoint by E. E. Moise (Addison-Wesley, 1963) an excellent reference.
Except I recently got this book and it does not do this. It is an interesting book, but it simply accepts the primitive notions ("the geometric terms") and Hilbert's axioms. I would like to see the construction of these entities directly from $\mathbb{R}^n$, Binmore defines lines, circles, points, and planes. But nowhere is there congruence (presumably for line segments this would be the usual distance between two points) or betweenness defined (presumably a point $b$ would be between $a$ and $c$ if $d(a,c) = d(a,b) + d(b,c))$. I've attempted to do this myself but I'm in over my head a little bit.
So, does anyone have resources that systematically defines each of these geometric objects as sets of $\mathbb{R}^n$ and then proves hilbert's axioms as theorems in $\mathbb{R}^n$? Especially Euclid's postulate.
What you have is basically real linear algebra with usual metric. Define a point as an element of $\Bbb R^n$, a line as the set $\vec{a} + t\vec{b}$, betweeness exactly like you did (using distances - in this case, the norm), and planes and other hyperplanes as you'd do with subspaces on $\Bbb R^n$.
You’ll might be interested in this question.