\(EF^2=AF^2+AE^2=\dfrac{9}{16}AD^2+\dfrac{1}{9}AB^2\)
\(CF^2=DF^2+CD^2=\dfrac{1}{16}AD^2+AB^2\)
\(CE^2=BC^2+EB^2=AD^2+\dfrac{4}{9}AB^2\)
Theo Pitago: \(EF^2+CF^2=CE^2\Rightarrow16AB^2=9AD^2\Rightarrow AD=\dfrac{4}{3}AB\)
\(\Rightarrow\left\{{}\begin{matrix}EF^2=\dfrac{10}{9}AB^2\\CF^2=\dfrac{10}{9}AB^2\end{matrix}\right.\) \(\Rightarrow EF=CF\)
Gọi H là hình chiếu vuông góc của F lên CE \(\Rightarrow H\) là trung điểm CE
Phương trình HF: \(3\left(x-2\right)+1\left(y-1\right)=0\Leftrightarrow3x+y-7=0\)
Tọa độ H là nghiệm: \(\left\{{}\begin{matrix}x-3y-9=0\\3x+y-7=0\end{matrix}\right.\) \(\Rightarrow H\left(3;-2\right)\)
Gọi \(C\left(3c+9;c\right)\Rightarrow E\left(-3c-3;-c-4\right)\) \(\Rightarrow\left\{{}\begin{matrix}\overrightarrow{EF}=\left(3c+5;c+5\right)\\\overrightarrow{FC}=\left(3c+7;c-1\right)\end{matrix}\right.\)
\(EF\perp CF\Rightarrow\left(3c+5\right)\left(3c+7\right)+\left(c+5\right)\left(c-1\right)=0\)
\(\Leftrightarrow c^2+4c+3=0\Rightarrow\left[{}\begin{matrix}c=-1\Rightarrow C\left(6;-1\right)\\c=-3\Rightarrow C\left(0;-3\right)\left(loại\right)\end{matrix}\right.\)
