Question 12
In the triangle $ABC$ we denote by $O$ and $I$ the circumcenter and the incenter, respectively. The perpendicular bisectors of the line segments $IA$, $IB$ and $IC$ pairwise intersect, thus defining the triangle $A_1B_1C_1$. Prove that
$$
\vec{OI} = \vec{OA_1} + \vec{OB_1} + \vec{OC_1}.
$$
$$
\vec{OI} = \vec{OA_1} + \vec{OB_1} + \vec{OC_1}.
$$