Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.
Školjka 
be a triangle with circumcircle 
and incenter 
and let 
be the midpoint of 
. The points 
, 
, 
are selected on sides 
, 
such that 
, 
, and 
. Suppose that the circumcircle of 
intersects 
other than 
. Prove that lines 
and 
meet on