Given Three points $A$, $B$ and $C$ where the horizontal distance $BC$ is known, as is the distance $AB - AC$, and the coordinates of point $A$ = $(A_x, A_y)$ are known, find the coordinates of point $C = (C_x,C_y)$.
The real world application of this problem is to assist in the calibration of a hanging plotter. To calibrate the hanging plotter, the initial position of the head must be known, however measuring the initial position relative to one of the motors ($A$) with accuracy can be difficult because the distance can be large and the measuring path can be obstructed. It would be much easier and more accurate to mark two points ($B$ and $C$) a known distance (say 1 foot) apart, and then manually move the head from one to the other. From this action the change in length of the control chain ($AB$ or $AC$) can be recorded, and the distance $AB - AC$ is known.
Please ask if parts of the question are unclear. I have done my best to describe the situation succinctly, but I may have left something out.
I am not sure that the problem has a single solution as described.
This is a work in progress. Any and all suggestions are welcome.
We have the known quantities: $$L_{BC}$$ $$L_{X} = L_{AB} - L_{AC}$$ $$A_X$$ $$A_y$$
And the known equations: $$B_y = C_y\tag1$$ $$B_x = C_x - L_{BC}\tag2$$ $$(B_x-A_x)^2+(B_y-A_y)^2={R_{AB}}^2\tag3$$ $$(C_x-A_x)^2+(C_y-A_y)^2={R_{AC}}^2\tag4$$ $$R_{AC} - R_{AB} = L_X \tag5$$
Plugging $(1)$ and $(2)$ into $(3)$ we find:
$$((C_x - L_{BC})-A_x)^2+(C_y-A_y)^2={R_{AB}}^2\tag6$$
Solving that equation for $R_{AB}$ and $(4)$ for $R_{AC}$ we find:
$$\sqrt{(C_x - L_{BC})-A_x)^2+(C_y-A_y)^2} = R_{AB}\tag6$$ $$\sqrt{(C_x-A_x)^2+(C_y-A_y)^2} = R_{AC}\tag4$$
Plunging these equations into $(5)$ we find:
$$\sqrt{(C_x-A_x)^2+(C_y-A_y)^2} - \sqrt{(C_x - L_{BC})-A_x)^2+(C_y-A_y)^2} = L_X \tag7$$
Which leaves us with a single equation and two variables $C_x, C_y$ so a single solution cannot be found from the given information.
For the sake visualizing what the solution space looks like, let's assume the values of $(A_x, A_y) = (0,0)$ leaving us with the equation:
$$\sqrt{C_x^2+C_y^2} - \sqrt{(C_x - L_{BC})^2+C_y^2} = L_X \tag7$$
Making up some values for $L_x$ and $L_{BC}$ as $L_{BC} = 100mm, L_x = 30mm$
We can see that the solution space is a parabola like so:
Hopefully applying the same logic to the other side of the machine will result in a similar parabola with a single intersection point with this parabola.