US-TST 2017 Problem 2

Let $A B C$ be an acute scalene triangle with circumcenter $O$, and let $T$ be on line $B C$ such that $\angle T A O=90^{\circ}$. The circle with diameter $\overline{A T}$ intersects the circumcircle of $\triangle B O C$ at two points $A_{1}$ and $A_{2}$, where $O A_{1}<O A_{2}$. Points $B_{1}, B_{2}, C_{1}, C_{2}$ are defined analogously. (a) Prove that $\overline{A A_{1}}, \overline{B B_{1}}, \overline{C C_{1}}$ are concurrent. (b) Prove that $\overline{A A_{2}}, \overline{B B_{2}}, \overline{C C_{2}}$ are concurrent on the Euler line of triangle $A B C$.