Ta thấy: \(S_{\Delta ADN}=\dfrac{AB\cdot AD}{2}\)
Lại thấy: \(S_{ADN}=S_{APD}+S_{DPQR}+S_{QRN}\)
\(\Rightarrow\dfrac{AB\cdot AD}{2}=S_{APD}+S_{DPQR}+S_{QRN}\)
\(\Rightarrow S_{DPQR}=\dfrac{AB\cdot AD}{2}-S_{APD}-S_{QRN}\)\(^{\left(1\right)}\)
Mặt khác: \(S_{CBM}+S_{ADM}=\dfrac{AD\cdot BM}{2}+\dfrac{AD\cdot AM}{2}\)
\(=\dfrac{AD\left(BM+AM\right)}{2}=\dfrac{AD\cdot AB}{2}\)
Lại thấy: \(S_{DAM}+S_{CBM}=S_{APD}+\left(S_{MAP}+S_{MQNB}+S_{RNC}\right)+S_{QRN}\)
\(\Rightarrow\dfrac{AD\cdot AB}{2}=S_{APD}+\left(S_{MAP}+S_{MQNB}+S_{RNC}\right)+S_{QRN}\)
\(\Rightarrow\left(S_{MAP}+S_{MQNB}+S_{RNC}\right)=\dfrac{AD\cdot AB}{2}-S_{APD}-S_{QRN}\)\(^{\left(2\right)}\)
Từ (1) và (2) \(\Rightarrow S_{DPQR}=\left(S_{MAP}+S_{MQNB}+S_{QRN}\right)\)
Mà \(S_{DPQR}=1\) (giả thiết)
\(\Rightarrow\left(S_{MAP}+S_{MQNB}+S_{QRN}\right)=1\left(đpcm\right)\)
