Question 22
In an acute triangle $A B C$ let $K, L$, and $M$ be the midpoints of sides $A B, B C$, and $C A$, respectively. From each of $K, L$, and $M$ drop two perpendiculars to the other two sides of the triangle; e.g., drop perpendiculars from $K$ to sides $B C$ and $C A$, etc. The resulting 6 perpendiculars intersect at points $Q, S$, and $T$ as in the figure to form a hexagon $K Q L S M T$ inside triangle $A B C$. Prove that the area of the hexagon $K Q L S M T$ is half of the area of the original triangle $A B C$.
