Question 33
The acute-angled triangle $ABC$ has circumcenter $O$. The straight lines $B_aC_a$, $C_bA_b$ and $A_cB_c$ are perpendicular in $O$ on the straight lines $AO$, $BO$, respectively $CO$ and $A_b$, $A_c \in BC$, $B_a, B_c \in AC$, $C_b, C_a \in AB$. Denote $O_a, O_b, O_c$ the circumcenters of the triangles $AC_aB_a, BA_bC_b, CB_cA_c$. Prove that
$$
\vec{O_aA} + \vec{O_bB} + \vec{O_cC} = \vec{0} .
$$
$$
\vec{O_aA} + \vec{O_bB} + \vec{O_cC} = \vec{0} .
$$