US-EGMO-TST 2019 Problem 5

Let the excircle of a triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$ . Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$ , respectively. Denote by $\gamma$ the circumcircle of triangle $A_1B_1C_1$ and assume that $\gamma$ passes through vertex $A$ . [list = a] [] Show that $\overline{AA_1}$ is a diameter of $\gamma$ . [] Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$ . [/list]