KöMaL 2009 A488

Let $P_1 P_2 P_3$ be a triangle with circumcenter $O$, the point $Q$ is in the triangle. Denote $t_i$ and $O_i$ the area and the circumcenter of the triangle $QP_{i+1} P_{i+2}$, respectively, where $i$=1,2,3 (the vertices are counted cyclically: $P_4$=$P_1$ and $P_5$=$P_2$). Prove that $t_1\cdot \overrightarrow{OO_1} + t_2\cdot \overrightarrow{OO_2} + t_3\cdot \overrightarrow{OO_3} = 0$.