A convex quadrilateral has an inscribed circle with center . Let and be the incenters of the triangles and , respectively. Suppose that the common external tangents of the circles and meet at , and the common external tangents of the circles and meet at . Prove that .
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 $AI_bI_d$ and $CI_bI_d$ meet at $X$, and the common external tangents of the circles $BI_aI_c$ and $DI_aI_c$ meet at $Y$. Prove that $\angle{XIY}=90^{\circ}$.
Školjka 
has an inscribed circle with center 
. Let 
and 
be the incenters of the triangles 
and 
, respectively. Suppose that the common external tangents of the circles 
and 
meet at 
, and the common external tangents of the circles 
and 
meet at 
. Prove that 
.