How to build a convex polygon from known vertex connections?

Question

Mathematics

There is some list of edges between vertices:
For example: [{A,B}, {B,C}, {C,A}]
It is known that the length of each edge of such a polygon must tend to some value L. That is, maybe more or less - but still somewhere in those areas)
So, it is required to find the coordinates of the vertices of such a polygon from this entry, which would be "maximally" convex (where possible). It is clear that there can be infinitely many solutions that satisfy such a condition. But given all the twists and turns, it's probably always the same. How to find it?
Recording examples:
[{A,B}, {B,C}, {C,A}] - an equilateral triangle with side length L
[{A,B}, {B,C}, {C,D}, {D ,A]] is a square with side length L.
[{A,B}, {B,C}, {C,D}, {D,E], {E,A}] - 5-gon with side length L.
[{A,B}, {B,C }, {C,D}, {D,E], {E,A}, {C,A}] - an equilateral triangle connected to a square along one of the edges.
You can take, build a physical model of the repulsion of charges with bonds in the form of springs, and iteratively come to a solution (which may not satisfy the initial conditions). What else can you think of here?