Let be a triangle, and its incenter. Consider a circle which lies inside the circumcircle of triangle and touches it, and which also touches the sides and of triangle at the points and , respectively. Show that the point is the midpoint of the segment .
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Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.
Source: Međunarodna matematička olimpijada, shortlist 1993
Školjka 
be a triangle, and 
its incenter. Consider a circle which lies inside the circumcircle of triangle 
and 
of triangle 
and 
, respectively. Show that the point 
.