Question 89
A convex quadrilateral $ABCD$ has an inscribed circle with center $I$. Let $I_{a}$, $I_{b}$, $I_{c}$, and $I_{d}$ be the incenters of the triangles $DAB$, $ABC$, $BCD$, and $CDA$, respectively. Suppose that the common external tangents of the circles $A I_{b} I_{d}$ and $C I_{b} I_{d}$ meet at $X$, and the common external tangents of the circles $B I_{a} I_{c}$ and $D I_{a} I_{c}$ meet at $Y$. Prove that $\angle X I Y = 90^{\circ}$.