The Euler line states that the orthocenter, circumcenter and centroid of a given triangle are on one line. This made me wondering whether the following is true:
For every three points on a line (not necessarily different), does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid?
I already eliminated the case that two points coincide and the third not, because if two points coincide, the the third point is also on that point. I assume that if two points are very close and the third is far away, it is not possible, but I'm not able to show it.
I thought of this problem myself.