Solving for points in a plane based on line lengths and geometry

Question

I have the following points and lines in a plane:

The problem is this:

Given that we know the lengths of lines A, B and C, how can we calculate the coordinates of each point a, b and c? The problem is set up to be solved with a system of non-linear equations. We can change the lengths of A, B and C to create the system. For simplicity, I'll say that the $X_1$ and $Y_1$ points are stationary and the $X_2$ and $Y_2$ points increase/decrease as the length of the line changes. Also, the distance between the $X_2$ and $Y_2$ points remain the same as the lengths change; assume it's a rigid structure. I am allowed to choose one known point in the middle of the plane to have a known value such as (1,1) to define the coordinate system. They are not equidistant from one another. If I am able to change the lengths of the lines, how would I turn this into a system of equations? Or if there's another, better method of solving this problem, please let me know.

To summarize --

Knowns:

• Lengths of A, B and C
• The $X_2,Y_2$ points remain the same distance from one another (not equidistant, but the distance remains the same from point to point as the lengths change).
• The Lengths of A, B and C can change as many times as needed to solve the system.
• I can choose one point in the plane to have a known value at e.g. the exact middle can be (1,1)

Let me know if any additional details are needed. Any help is much appreciated!

Answer 1

Here's a little sidenote: the length of the lines cannot change. The way you are construing the problem is a bit deceptive because you are drawing in a square to represent three the relationship between points, but to better represent these distances, we should represent it as a triangle.

By stating that the distance between points cannot change, you are saying that the lengths of the lines can only change if the end product produces a triangle that is congruent to the current triangle, by SSS congruence. The only way to do that would be to either rotate the triangle (which is not allowed because the slopes can't change) or move the triangle (which is not allowed because the lines can't move).

Also, the one point in the center is absolutely useless without giving reference to another location on the actual object. Scale cannot be assumed, so the distance between (0,0) and BX2 and BY2 could be 1 and it could be 1000.

As far as I can see, this is just not enough info.

Answer 2

Imagine a number line with points A, B, and C. A is 3 units less than B and C is 3 units more than C. Something like

----------------------------------------------------------------------
           A    '    '    B    '    '    C

We are also told that way over there, in the distance, is the number 27.

---------------------------------------------------------------------------
           A    '    '    B    '    '    C                              27

We don't know how 27 relates to any of the other numbers, but it is just there. Do you see how it might be possible to tell the relationships between the variables but impossible to discover the actual variables themselves?

With some random point in the center of the graph, we can't really relate it to the rest of the drawing. However, it is possible to tell the distance between A and B as well as between B and C, and even between C and A, but without any definitive numbers, it is impossible to figure out any definite solutions for the variables.