\(\left\{{}\begin{matrix}\overrightarrow{AM}=\left(x-1;y-3\right)\\\overrightarrow{BM}=\left(x-4;y-2\right)\end{matrix}\right.\)
Tam giác ABM vuông cân tại M khi:
\(\left\{{}\begin{matrix}\overrightarrow{AM}.\overrightarrow{BM}=0\\AM^2=BM^2\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}\left(x-1\right)\left(x-4\right)+\left(y-3\right)\left(y-2\right)=0\\\left(x-1\right)^2+\left(y-3\right)^2=\left(x-4\right)^2+\left(y-2\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2-5x+y^2-5y+10=0\\3x-y=5\end{matrix}\right.\)
Thế \(y=3x-5\) lên pt trên:
\(x^2-5x+\left(3x-5\right)^2-5\left(3x-5\right)+10=0\)
\(\Leftrightarrow x^2-5x+6=0\Rightarrow\left[{}\begin{matrix}x=2\Rightarrow y=1\\x=3\Rightarrow y=4\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}M\left(2;1\right)\\M\left(3;4\right)\end{matrix}\right.\)
