While working with a student today, I had the unfortunate realization that I do not actually know a rigourous definition for a vertex (here in the sense of plane geometry). It didn't stop us from working, thankfully, but it has been nagging at me since.
What is a vertex of a polygon?
I think a definition that satisfies me would be:
"Clearly" able to be phrased in ZFC, and not an alternate axiomatic system for the purposes of plane geometry (sorry Hilbert)
Able to be "worked with", though not necessarily be easy or intuitive to work with.
Capture various "intuitive" notions of vertices (that they are pieces of a polygon, that they are formed by straight edges, that a hexagon has six), while avoiding including "unnatural" vertices (in particular, I would not like to consider an intersection of two curves to be a vertex).
One option I considered was "the image of the convex hull of the $n$th roots of unity under a homeomorphism which preserves the linearity of the edges" is a polygon, and then the images of the roots of unity are the vertices, but I am not overly satisfied by this. What are some alternate definitions?