Is it possible to determine triangle with prescribed centres (incentre, orthocentre, barycentre etc.)

Question

The centres of a triangle is related to the triangle itself, or in the language of coordinate geometry, their coordinates can be calculated from that of the triangle's vertices. Can we reverse this process? More specifically, to what extent is it possible and to what extent are we free to choose the centres? (e.g. if the barycentre, the orthocentre and the cincumcentre do not follow Euler's theorem it cannot be from a triangle) If we can is the determined triangle unique?

Answer 1

In a triangle, we have 6 independent parameters. In case the geometry center is given, it reduces to 4. Sure, if 3 independently defined "centers" are provided, then, inversely, the triangle is defined.