In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
Školjka 
, let 
be the excircle opposite to 
. Let 
and 
be the points where 
, and 
, respectively. The circle 
intersects line 
at 
and 
. Let 
be the midpoint of 
. Prove that the circle 
is tangent to