8. Let $A B C D E F$ be a regular hexagon of area 1 . Let $M$ be the midpoint of $D E$. Let $X$ be the intersection of $A C$ and $B M$, let $Y$ be the intersection of $B F$ and $A M$, and let $Z$ be the intersection of $A C$ and $B F$. If $[P]$ denotes the area of polygon $P$ for any polygon $P$ in the plane, evaluate $[B X C]+[A Y F]+[A B Z]-[M X Z Y]$.
Answer: $\square$
Let $O$ be the center of the hexagon. The desired area is $[A B C D E F]-[A C D M]-[B F E M]$. Note that $[A D M]=[A D E] / 2=[O D E]=[A B C]$, where the last equation holds because $\sin 60^{\circ}=\sin 120^{\circ}$. Thus, $[A C D M]=[A C D]+[A D M]=[A C D]+[A B C]=[A B C D]$, but the area of $A B C D$ is half the area of the hexagon. Similarly, the area of $[B F E M]$ is half the area of the hexagon, so the answer is zero.