Đáp án:
$\begin{array}{l}
a)E\left( {x;y} \right)\\
\Rightarrow \overrightarrow {EA} = \left( {3 - x; - 1 - y} \right)\\
\overrightarrow {EB} = \left( {6 - x; - y} \right)\\
\overrightarrow {AC} = \left( { - 2;6} \right)\\
Do:\overrightarrow {EA} - 2\overrightarrow {EB} = 3\overrightarrow {AC} \\
\Rightarrow \left\{ \begin{array}{l}
3 - x - 2\left( {6 - x} \right) = 3.\left( { - 2} \right)\\
- 1 - y - 2\left( { - y} \right) = 3.6
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
x = 3\\
y = 19
\end{array} \right.\\
\Rightarrow E\left( {3;19} \right)\\
b)\left\{ \begin{array}{l}
{x_G} = \frac{{3 + 6 + 1}}{3} = \frac{{10}}{3}\\
{y_G} = \frac{{ - 1 + 5}}{3} = \frac{4}{3}
\end{array} \right. \Rightarrow G\left( {\frac{{10}}{3};\frac{4}{3}} \right)\\
c)F\left( {x;0} \right)\\
C{B^2} = C{F^2}\\
\Rightarrow {\left( {6 - 1} \right)^2} + {5^2} = {\left( {x - 1} \right)^2} + {5^2}\\
\Rightarrow 50 = {\left( {x - 1} \right)^2} + {5^2}\\
\Rightarrow x - 1 = - 5\\
\Rightarrow x = - 4\\
\Rightarrow F\left( { - 4;0} \right)\\
d)BC:y = - x + 6\\
\Rightarrow M\left( {x; - x + 6} \right)\\
\Rightarrow AM = \sqrt {{{\left( {x - 3} \right)}^2} + {{\left( { - x + 6 + 1} \right)}^2}} \\
\Rightarrow A{M^2} = 2{x^2} - 20x + 58 = 2\left( {{x^2} - 10x + 25} \right) + 8 \ge 8\\
\Rightarrow AM \ge 2\sqrt 2 \\
Dấu = xảy\,ra \Leftrightarrow x = 5\\
\Rightarrow M\left( {5;1} \right)
\end{array}$
