US-TST 2019 Problem 6

Let $A B C$ be a triangle with incenter $I$, and let $D$ be a point on line $B C$ satisfying $\angle A I D=90^{\circ}$. Let the excircle of triangle $A B C$ opposite the vertex $A$ be tangent to $\overline{B C}$ at point $A_{1}$. Define points $B_{1}$ on $\overline{C A}$ and $C_{1}$ on $\overline{A B}$ analogously, using the excircles opposite $B$ and $C$, respectively. Prove that if quadrilateral $A B_{1} A_{1} C_{1}$ is cyclic, then $\overline{A D}$ is tangent to the circumcircle of $\triangle D B_{1} C_{1}$.