Question 10
A circle is circumscribed about the triangle $ABC$. $X$ is the midpoint of the arc $BC$ (on the opposite side of $BC$ to $A$), $Y$ is the midpoint of the arc $AC$, and $Z$ is the midpoint of the arc $AB$. $YZ$ meets $AB$ at $D$ and $YX$ meets $BC$ at $E$. Prove that $DE$ is parallel to $AC$ and that $DE$ passes through the center of the inscribed circle of $ABC$.