\[\begin{array}{l}
a)\,\,Goi\,\,\,D\left( {a;\,\,b} \right).\\
B\,\,la\,\,trong\,\,tam\,\,\Delta ACD \Rightarrow \left\{ \begin{array}{l}
{x_B} = \frac{{{x_A} + {x_C} + {x_D}}}{3}\\
{y_B} = \frac{{{y_A} + {y_C} + {y_D}}}{3}
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
3 = \frac{{2 - 4 + a}}{3}\\
5 = \frac{{3 + 1 + b}}{3}
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = 11\\
b = 11
\end{array} \right. \Rightarrow D\left( {11;\,\,11} \right).\\
b)\,\,\,M\,\,la\,\,trung\,\,diem\,\,cua\,\,BC \Rightarrow M\left( { - \frac{1}{2};\,\,3} \right).\\
N\,\,la\,\,\,trung\,\,diem\,\,cua\,\,AM \Rightarrow N\left( {\frac{3}{4};\,\,3} \right).\\
Phuong\,\,\,trinh\,\,duong\,\,thang\,\,di\,\,qua\,\,A,\,\,C\,\,la:\\
\frac{{x - 2}}{{ - 4 - 2}} = \frac{{y - 3}}{{1 - 3}} \Leftrightarrow x - 2 = 3\left( {y - 3} \right) \Leftrightarrow x - 3y + 7 = 0\\
Phuong\,\,trinh\,\,duong\,\,thang\,\,di\,\,qua\,\,B\,\,\,va\,\,\,N\,\,la:\\
\frac{{x - 3}}{{\frac{3}{4} - 3}} = \frac{{y - 5}}{{3 - 5}} \Leftrightarrow - 2\left( {x - 3} \right) = - \frac{9}{4}\left( {y - 5} \right)\\
\Leftrightarrow 8x - 24 = 9y - 45 \Leftrightarrow 8x - 9y + 21 = 0\\
E = \left\{ {AC} \right\} \cap BN \Rightarrow toa\,\,\,do\,\,diem\,\,E\,\,la\,\,nghiem\,\,cua\,\,\,hpt:\\
\left\{ \begin{array}{l}
x - 3y + 7 = 0\\
8x - 9y + 21 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = \frac{{10}}{3}\\
y = \frac{{31}}{9}
\end{array} \right. \Rightarrow E\left( {0;\,\,\frac{7}{3}} \right).
\end{array}\]
