Trong mp (ABCD), qua D kẻ đường thẳng song song MI cắt AB tại F
Trong mp (SAB), qua F kẻ đường thẳng song song SA cắt SB tại K
\(\overrightarrow{MD}=-2\overrightarrow{MC}=-2\left(\overrightarrow{MD}+\overrightarrow{DC}\right)\Rightarrow\overrightarrow{MD}=\dfrac{2}{3}\overrightarrow{CD}\) và \(\overrightarrow{MC}=\dfrac{1}{3}\overrightarrow{DC}=\dfrac{1}{2}\overrightarrow{AB}\)
\(\Rightarrow\overrightarrow{MD}=\overrightarrow{BA}\Rightarrow ABMD\) là hình vuông \(\Rightarrow\overrightarrow{BM}=\overrightarrow{AD}\)
\(2\overrightarrow{BE}=\overrightarrow{EM}=\overrightarrow{EB}+\overrightarrow{BM}\Rightarrow\overrightarrow{BE}=\dfrac{1}{3}\overrightarrow{BM}=\dfrac{1}{3}\overrightarrow{AD}\)
Đặt \(\overrightarrow{BI}=x.\overrightarrow{BC}=x.\left(\overrightarrow{BM}+\overrightarrow{MC}\right)=x\left(\overrightarrow{AD}+\dfrac{1}{2}\overrightarrow{AB}\right)=x.\overrightarrow{AD}+\dfrac{x}{2}.\overrightarrow{AB}\)
\(\overrightarrow{AE}=\overrightarrow{AB}+\overrightarrow{BE}=\overrightarrow{AB}+\dfrac{1}{3}\overrightarrow{AD}\)
\(\overrightarrow{AI}=\overrightarrow{AB}+\overrightarrow{BI}=\overrightarrow{AB}+x.\overrightarrow{AD}+\dfrac{x}{2}\overrightarrow{AB}=\left(1+\dfrac{x}{2}\right)\overrightarrow{AB}+x.\overrightarrow{AD}\)
A, E, I thẳng hàng \(\Rightarrow\dfrac{1+\dfrac{x}{2}}{1}=3x\Rightarrow x=\dfrac{2}{5}\)
\(\Rightarrow\overrightarrow{MI}=\overrightarrow{MB}+\overrightarrow{BI}=-\overrightarrow{AD}+\dfrac{2}{5}\overrightarrow{AD}+\dfrac{1}{5}\overrightarrow{AB}=\dfrac{1}{5}\overrightarrow{AB}-\dfrac{3}{5}\overrightarrow{AD}\)
Đặt \(\overrightarrow{AF}=y.\overrightarrow{AB}\Rightarrow\overrightarrow{DF}=\overrightarrow{DA}+\overrightarrow{AF}=y.\overrightarrow{AB}-\overrightarrow{AD}\)
DF song song MI \(\Rightarrow\dfrac{y}{\dfrac{1}{5}}=\dfrac{-1}{-\dfrac{3}{5}}\Rightarrow y=\dfrac{1}{3}\)
Theo định lý Talet: \(\dfrac{SK}{SB}=\dfrac{AF}{AB}=y=\dfrac{1}{3}\)
