In the triangle $ABC\,$ we denote by $O$ and $I$ the circumcenter and 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 $\vec {OI}=\vec {OA_1}+\vec {OB_1}+\vec {OC_1}$
Hi there. Im a high schooler new to olympiad geometry. Ive been unable to solve this problem despite repeated angle chasing could someone give a solution?Note I dont know stuff like projective geometry and hard lemmas so a simple proof would be most appreciated.
In another words, it is enough to prove that $O$ is circumcenter of $A_1B_1C_1$ and $I$ is it orthocenter (and this is so called Hamilton's theorem).
And this is easy to prove. Say $AI,BI,CI$ cuts circumcircle of $ABC$ at $A',B'C'$ respectively. Then $A'B'$ is perpendicular bisector of $CI$ (can you prove it your self?) and so on and $A',B',C'$ are intersection points of those bisectors.