OBM 2017 Problem 5

In triangle $ABC$ , let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$ . Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$ , respectively. Suppose that the three lines $r_A$ , $r_B$ , $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$ .