Folklore

Points $K$ and $L$ are the midpoints of sides $A B$ and $B C$ of a regular hexagon $A B C D E F$. Segments $K D$ and $L E$ intersect at point $M$. The area of triangle $D E M$ is 12. Find the area of quadrilateral KBLM.
Since quadrilateral $K B C D$ is the image of quadrilateral $L C D E$ under a rotation around the center of $A B C D E F$ by an angle of $60^{\circ}$, these quadrilaterals are equal, and therefore, have the same area (see figure). By subtracting the area of quadrilateral $L C D M$ from each of them, we get that $S_{K B L M}=S_{D E M}=12$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b324abf723c207e3278g-28.jpg?height=351&width=437&top_left_y=1271&top_left_x=816)

Answer

12.