By relative coordinate system I am refering to a specific system where each coordinate is given only as its distance away from other points (if this kind of system has a more accurate name then please let me know as I am struggling to find any information on a system like it), I am fairly certain this would require at least three distances per point as with only two distances there would be two intersection points, with the third you could narrow that down to a single point. However I welcome being corrected on that assumption.
The problem I am specifically asking about is what happens when you need to calculate the distance between two points which do not have a direct link. In this case I would assume every distance other than the distance we are trying to find is already known.
To ask about this in a way that can be explained lets consider a quadralateral network ABCD where the distance between all of the nodes is known. Then a node E is added and AE, BE, and CE are defined.
Can the distance DE be calculated just using the information given?
I thought I had found an answer with this (How can I find the diagonal of a quadrilateral?) question, but the answers assume that the quadralateral is convex and I don't think that assumption can be made.
You should establish a coordinate system yourself.
Start with two points $A, B$. Choose $A$ to be the origin. It has coordinates $(0,0)$. $B$ will define the direction of the $x$-axis. So the coordinates of $B$ are $(b,0)$, where $b = d(A, B)$ is the distance between $A$ and $B$. $(1,0)$ is the point at a distance of $1$ away from $A$ in the direction of $B$.
You will need a third point $C$, not in line with $A$ and $B$, to define the positive-$y$ side of the $x$-axis. Suppose $C = (x,y)$ is at a distance $r_a = d(A, C), r_b = d(B, C)$ from $A$ and $B$. This gives a system of two equations $$x^2 + y^2 = r_a^2\\(x - b)^2 + y^2 = r_b^2$$ If you expand the second equation and then subtract it from the first and simplify, you get: $$x = \frac{r_a^2 - r_b^2 + b^2}{2b}$$ Since we've chosen $C$ to be in the positive $y$ side of the plane, we have $y = \sqrt{r_a^2 - x^2}$.
Now for all future points you can calculate their coordinates by similarly solving the equations for their distance from two existing points, and as you've described, using a third point to differentiate between the two solutions. Once you know the coordinates, you can just calculate distance to any other known point.