Question 69
Let $ABC$ be an acute, non-isosceles triangle with $M, N, P$ as the midpoints of $BC, CA, AB$, respectively. Denote $d_{1}$ as the line passing through $M$ and perpendicular to the angle bisector of $\angle BAC$, and similarly define $d_{2}, d_{3}$. Suppose that $d_{2} \cap d_{3} = D$, $d_{3} \cap d_{1} = E$, $d_{1} \cap d_{2} = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.