Question 63
A convex quadrilateral $ABCD$ has an inscribed circle with centre $I$. Let $I_a$, $I_b$, $I_c$ and $I_d$ be the centres of the inscribed circles of the triangles $DAB$, $ABC$, $BCD$ and $CDA$ respectively. Suppose that the common tangents to the circumcircles of $AI_bI_d$ and $CI_bI_d$ meet at $X$ and that the external common tangents of the circumcircles of $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\widehat{XIY} = 90^\circ$.