MathNet 2014 Question 85

Let $U$ be the circumcenter of the acute-angled triangle $\triangle ABC$. Furthermore, let $M_A$, $M_B$ and $M_C$ be the circumcenters of the triangles $\triangle UBC$, $\triangle UAC$ and $\triangle UAB$ in this order. For which triangles $\triangle ABC$ is the triangle $\triangle M_A M_B M_C$ similar to the original triangle (independent of the order of the vertices)?
G. Baron, Vienna