Problem

Source: USA TST for EGMO 2019, Problem 5 (adapted from IMO TST Problem 6)

Tags: geometry, Weird, USA



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 $A$. Show that $\overline{AA_1}$ is a diameter of $\gamma$. Show that the incenter of $\triangle ABC$ lies on line $B_1C_1$.