Xét \(\Delta ADN\) có:
\(\left\{{}\begin{matrix}S_{ADN}=\dfrac{AB\cdot AD}{2}\\S_{ADN}=S_{ADP}+S_{DPQR}+S_{QRN}\end{matrix}\right.\)
\(\Rightarrow\dfrac{AB\cdot AD}{2}=S_{ADP}+S_{DPQR}+S_{QRN}\)
\(\Rightarrow S_{DPQR}=\dfrac{AB\cdot AC}{2}-S_{ADP}-S_{QRN}\)(1)
Xét \(\Delta BCM\) và \(\Delta ADM\) có:
\(\left\{{}\begin{matrix}S_{BCM}+S_{ADM}=\dfrac{AM\cdot AD}{2}+\dfrac{BM\cdot AD}{2}\\S_{BCM}+S_{ADM}=\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)+S_{PAD}+S_{NRQ}\end{matrix}\right.\)
\(\Rightarrow\) \(\left\{{}\begin{matrix}S_{BCM}+S_{ADM}=\dfrac{\left(AM+BM\right)BC}{2}\\S_{BCM}+S_{ADM}=\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)+S_{PAD}+S_{NRQ}\end{matrix}\right.\)
\(\Rightarrow\) \(\left\{{}\begin{matrix}S_{BCM}+S_{ADM}=\dfrac{AB\cdot AD}{2}\\S_{BCM}+S_{ADM}=\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)+S_{PAD}+S_{NRQ}\end{matrix}\right.\)
\(\Rightarrow\dfrac{AB\cdot AD}{2}=\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)+S_{PAD}+S_{NRQ}\)
\(\)\(\Rightarrow\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)=\dfrac{AB\cdot AD}{2}- S_{PAD}-S_{NRQ}\) (2)
Từ (1) và (2) \(\Rightarrow S_{DPQR}=\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)\)
Mà \(S_{DPQR}=1\left(gt\right)\)
\(\Rightarrow\left(S_{MPA}+S_{NQMB}+S_{NCR}\right)=1\) (đpcm)
