SHARYGIN 2018 Problem 12

Let $BD$ be the external bisector of a triangle $ABC$ with $AB > BC$ ; $K$ and $K_1$ be the touching points of side $AC$ with the incircle and the excircle centered at $I$ and $I_1$ respectively. The lines $BK$ and $DI_1$ meet at point $X$ , and the lines $BK_1$ and $DI$ meet at point $Y$ . Prove that $XY \perp AC$ .