MathNet 2018-part-03 — 181 problems (EN–VI) Question 53

Points $X$ and $Y$ respectively lie on the tangent lines to the circumcircle of triangle $ABC$ passing through $B$, $C$ such that $AB = BX$ and $AC = CY$. (Points $X$, $Y$, $A$ are on the same side of line $BC$.) Let $I$ be the incenter of triangle $ABC$, prove that
$$
\widehat{BAC} + \widehat{XIY} = 180^\circ
$$