It is well-known that for an equilateral triangle the centroid, incenter and circumcenter are all the same, and that for any non-equilateral triangle these three centers are different points. So, apart from equilateral (degenerate) case, can we find the vertices of a triangle given its centroid, incenter and circumcenter?
It seems to me that the answer is positive, though I couldn't find literature on these question.
Note: The case of isosceles triangles is rather easy, since the incenter lies in the Euler line and the problem can be transformed into a one dimensional equation.