\(S_{AMD}=\frac{1}{2}\times S_{ABD}\)(chung đường cao hạ từ \(D\), \(AM=\frac{1}{2}\times AB\))
\(S_{AMQ}=\frac{1}{2}\times S_{AMD}\)(chung đường cao hạ từ \(M\), \(AQ=\frac{1}{2}\times AD\))
Suy ra \(S_{AMQ}=\frac{1}{2}\times\frac{1}{2}\times S_{ABD}=\frac{1}{4}\times S_{ABD}\)
Tương tự ta cũng có: \(S_{BMN}=\frac{1}{4}\times S_{BAC},S_{CNP}=\frac{1}{4}\times S_{CBD},S_{DPQ}=\frac{1}{4}\times S_{DAC}\)
Suy ra \(S_{AMQ}+S_{BMN}+S_{CNP}+S_{DPQ}=\frac{1}{4}\times\left(S_{ABD}+S_{BAC}+S_{CBD}+S_{DAC}\right)\)
\(=\frac{1}{4}\times\left[\left(S_{ABD}+S_{CBD}\right)+\left(S_{BAC}+S_{DAC}\right)\right]\)
\(=\frac{1}{4}\times\left(S_{ABCD}+S_{ABCD}\right)=\frac{1}{2}\times S_{ABCD}\)
Suy ra \(S_{MNPQ}=S_{ABCD}-\left(S_{AMQ}+S_{BMN}+S_{CNP}+S_{DPQ}\right)=S_{ABCD}-\frac{1}{2}\times S_{ABCD}=\frac{1}{2}\times S_{ABCD}\)
