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I came across this problem by trying to construct a "nice" triangle, e.g. for an illustration, from 3 distances $\lbrace\|A-O\|, \|B-O\|, \|C-O\|\rbrace$ and the fuzzy requirement that $O$ be somewhere near the "center" of the triangle.

From that nebulous idea I distilled the following general

Question:

  • for which of the named triangle centers in Kimberling's Encyclopedia of Triangle Centers can triangles be constructed for which the center has a given triplet of distances to the corners
  • is it specifically possible for the incircle-center and for the centroid
  • for which triangle centers is the inverse problem solvable with straight-edge and compass

For the circum circle all distances must be equal, therefore in general no solution to its inverse problem exists.

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    $\begingroup$ This may not be what you are looking for, but at least it might help to get some empirical data. Using the $p,q$ method, i.e., assuming the triangle has vertices $(0,0)$,$(e_{12},0)$ and $(p,q)$ one can set up a Mathematica programme to compute the centre for any centre function. If you are given any data set, then this can be used to translate into three equations in $p,q,e_{12}$. I have tried this for many centres and given lengths and Mathematica always provided the solutions (the non-existence is protocolled by complex solutions). $\endgroup$ Commented Nov 27 at 14:28
  • $\begingroup$ The constructability condition is more delicate and will take more time. I have only done it for the (trivial, alas) case of the centroid. $\endgroup$ Commented Nov 27 at 14:33
  • $\begingroup$ @terceira even if it doesn't answer my question, it is an encouraging start... $\endgroup$ Commented Nov 27 at 15:24

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