Question 147
Let $ABC$ be an acute, non-isosceles triangle with circumcenter $O$, orthocenter $H$. Denote $d$ as any diameter of $(O)$ and $A'$ as the midpoint of $BC$. Let $A_1$ be the projection of $A$ on $d$, $A_2$ be the reflection of $A_1$ over the perpendicular bisector of $BC$ and $A_3$ be the midpoint of $A'A_2$. Define $B_3, C_3$ similarly. Prove that $ABC$ and $A_3B_3C_3$ are similar and find the ratio of similarity.