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I know I can construct a triangle with sides $(3,4,5)$. I also know that I cannot construct the triangle with sides (3,4,8), because the third side is longer than the sum of the two other.

Now let's say, I want to construct a geometrical figure with $n$ vertices with a given distance $d_{ij}$ between them vertices $i$ and $j$ (this geometrical figure could be described by a graph).

Is there a way to prove that this is or not possible to construct such a figure?

Is there an area in graph theory that is applied to determine the feasibility of a geometrical construction?

Ivan
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  • Wikipedia's "geometric graph theory" entry may give you a place to start. – Blue Apr 12 '20 at 02:32
  • If the size of the largest segment is greater than the sum of all the rest, than you can not construct a polygon. In all other cases you can and many of them for each set. – Moti Apr 12 '20 at 07:06
  • Imagine that I want to add a vertice to the (3,4,5) triangle so that it is a distance 1 from 2 of the vertices. This is not possible and the largest segment is not greater than the sum of the rest. I don't know if this is out of the definition of a polygon, but am looking for a more general answer. – Ivan Apr 13 '20 at 01:55
  • I think this thread answers your question. – Alex Ravsky Jul 03 '20 at 03:43

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