Sơ đồ minh họa:
Ta có:
\(S_{AED}=\frac{1}{2}\times AD\times AE=\frac{1}{2}\times AD\times\left(\frac{1}{4}\times AB\right)\)
\(=\frac{1}{8}\times AD\times AB=\frac{1}{8}\times S_{ABCD}\)
\(S_{BEF}=\frac{1}{2}\times BE\times BF=\frac{1}{2}\times\left(\frac{3}{4}\times AB\right)\times\left(\frac{1}{4}\times BC\right)\)
\(=\frac{3}{32}\times AB\times BC=\frac{3}{32}\times S_{ABCD}\)
\(S_{CDF}=\frac{1}{2}\times CD\times CF=\frac{1}{2}\times CD\times\left(\frac{3}{4}\times CB\right)\)
\(=\frac{3}{8}\times CD\times CB=\frac{3}{8}\times S_{ABCD}\)
Do đó: \(S_{AED}+S_{BEF}+S_{CDF}=\)
\(=\left(\frac{1}{8}+\frac{3}{32}+\frac{3}{8}\right)\times S_{ABCD}\)
\(=\frac{19}{32}\times S_{ABCD}\)
Suy ra:
\(S_{DEF}=S_{ABCD}-\left(S_{AED}+S_{BEF}+S_{CDF}\right)\)
\(=S_{ABCD}-\frac{19}{32}\times S_{ABCD}=\frac{13}{32}\times S_{ABCD}\)
Vậy \(\frac{S_{DEF}}{S_{ABCD}}=\frac{13}{32}\)
