Question 47
Let $I$ be the incenter of a triangle $ABC$. Let the incircle of $ABC$ be tangent to $CA$ and $AB$ at $E$ and $F$, respectively. Lines $BI$ and $CI$ intersect line $EF$ at $Y$ and $Z$, respectively. Denote by $M$, $N$ midpoints of segments $BC$ and $YZ$, respectively. Prove that $MN$ is parallel to $AI$.
Figure 22
Figure 22