I don't understand the assignment. Write an example for the specified figure, I'm interested in the structure of the source data. Structure = how you connect this data with each other. Is there an indication of which dimensions refer to one triangle or to adjacent triangles?
perhaps it is simpler if I explain the practical problem 
I have made a survey of a house .
"survey " means that I have made a SKETCH
and I have measured the
sides and
diagonals of each
room.
This is because there are some rooms that have an irregular shape.
I need to draw the room. . . .
Given two points A and B,
we need to determine the position of point C,
knowing the distances of C, from A and B.
This is the basic problem. Which is very simple. The problem is to determine the
complete shape of the polygon,
taking into account the available measures.
The measures can be insufficient,
sufficient or
overabundant.
The solution does not exist for an insufficient number of measures.
While, since the measures detected,
NOT PERFECTLY EXACT,
it is necessary to calculate as many times as the whole figure,
how many overabundant measures are there.
(but I think several times,
because each superabundant measure generates
a new series of combinations of known measures)
And a criterion must be found to calculate an average of the results.
For example, the same point C, calculating it using different or partially different data,
could result in a different position for each calculation.
And these results must be managed in order to find a "MIDDLE POSITION" of the point.
It is an ARCHITECTURAL SURVEY problem of ROOMS that have a shape
consisting of an irregular polygon whose sides (but not always all) and diagonals (some or all) are known.
It is not easy to explain it well.
In Italy, there is a software called ARTEN DIGIPLAN
http://www.artensrl.com/shop/it/softwarearten/4artendigiplan.htmlwhich solves this problem.
And certainly there are many others.
But it is complex.
I hope I managed to make myself understood a little better
I think we need to:
 define all possible triangles using known measurements.
 Calculate all these triangles
 Find a way to place them and connecting them each other
 Identify a criterion and a method to calculate the "average" position
of the vertices calculated many times
 and then connect all the external vertices, in order to obtain the final shape.
. . .
Be aware that the image I posted is a
sketch and
not a definite drawing.so the stages are:
 make a sketch
 take all possible measures

calculate and
draw the final shape