Question 29
Let the convex quadrilateral $ABCD$ have an incircle with center $I$. Let $I_a, I_b, I_c, I_d$ be the incenters of $\triangle DAB, \triangle ABC, \triangle BCD$ and $\triangle CDA$, respectively. Suppose that the two external common tangents of the circumcircles of $\triangle AI_bI_d$ and $\triangle CI_bI_d$ meet at the point $X$, and the two external common tangents of the circumcircles of $\triangle BI_aI_c$ and $\triangle DI_aI_c$ meet at the point $Y$.
Prove that $\angle XIY$ is a right angle.
Prove that $\angle XIY$ is a right angle.