9. Parallelogram $A E C F$ is inscribed in square $A B C D$. It is reflected across diagonal $A C$ to form another parallelogram $A E^{\prime} C F^{\prime}$. The region common to both parallelograms has area $m$ and perimeter $n$. Compute the value of $\frac{m}{n^{2}}$ if $A F: A D=1: 4$.
Solution: By symmetry, the region is a rhombus, AXCY, centered at the center of the square, $O$. Consider isoceles right triangle $A C D$. By the technique of mass points, we find that $D O: Y O=7: 1$. Therefore, the rhombus is composed of four triangles, whose sides are in the ratio $1: 7: 5 \sqrt{2}$. The perimeter of the rhombus is $20 \sqrt{2} N$, and the area is $14 N^{2}$. The required ratio is thus $\frac{7}{400}$.