But let's be more careful. Position is actually a relative quantity. Really, we should only ever write the position of two points relative to each other. We'll use e.g. ApC to denote the position of C measured from A. The left superscript looks mighty strange, but we'll see that it pays off once we start transforming points.
\(\renewcommand{\vec}[1]{\overrightarrow{#1}}\)While I appreciate the introduction of monogram notation and the algebraic rules given in section 3.3 below, I believe a small minority of readers (myself included) would appreciate a reference to a more formal introduction (e.g. an introduction that relates points to the concepts of vector spaces).
As far as I understand, the references provided in the introduction above do not formalize the notions given here, but instead develop them in more detail.
To clarify, the concept of a "point" (and its relationship to vectors in a vector space) is not rigorously defined here. Similarly, the addition operation of points "\({}^Ap^B_F + {}^Bp^C_F = {}^Ap^C_F.\)" given in \((1)\) came out of thin air. These are just two of the several concepts presented on this page that I think would benefit from more formal grounding.
Because of this, I looked around for a more formal overview and found [1] to be very helpful ([1] is freely available here from my end). In [1], the concept of an affine space is introduced (see also here).
After introducing the concept of an affine space, the author then shows that the addition operation "\({}^Ap^B_F + {}^Bp^C_F = {}^Ap^C_F\)" given in \((1)\) is nothing but Chasles' relation, which the author formally proves.
Additionally, the author explains affine combinations, affine maps, and other useful concepts (all of these concepts explain where the rest of the algebraic rules in section 3.3 came from). I believe this reference presents a different perspective that some readers may find helpful, and perhaps it can be mentioned somewhere near here.
References
[1] J. Gallier, “Basics of Affine Geometry,” in Geometric Methods and Applications: For Computer Science and Engineering, J. Gallier, Ed., New York, NY: Springer, 2011, pp. 7–63. doi: 10.1007/978-1-4419-9961-0_2.