Problem: On a given equilateral triangle \(ABC\), suppose that the points \(D\) and \(E\) are randomly selected such that the line segment \(\left[DE\right]\) divides \(ABC\) into two regions with equal area.
Prove that the ratio of the area of the region enclosed by the loci of midpoints of \(\left[DE\right]\) to the area of \(ABC\) is \(\ln\sqrt{\frac{2\sqrt{2}}{e}}\).