Ta có : \(\overrightarrow{n_{AH}}=\left(3;1\right)\Rightarrow\overrightarrow{u_{AH}}=\overrightarrow{n_{BC}}=\left(1;-3\right)\)
PTTQ BC đi qua điểm B và nhân \(\overrightarrow{n_{BC}}\) làm VTPT :
\(1\left(x-2\right)-3\left(y+7\right)=0\)
\(\Leftrightarrow x-3y-23=0\)
Gọi \(M\left(a;b\right)\) . Vì \(M\in CM\Rightarrow a+2b+7=0\Rightarrow b=\frac{-a-7}{2}\) . Do đó \(M\left(a;\frac{-a-7}{2}\right)\)
\(\Rightarrow\left\{{}\begin{matrix}x_A=2x_M-x_B=2a-2\\y_A=2y_M-y_B=-a\end{matrix}\right.\)
Vì \(A\in AH\) \(\Rightarrow3\left(2a-2\right)-a+11=0\) \(\Leftrightarrow a=-1\)
\(\Rightarrow A\left(-4;1\right);M\left(-1;-3\right)\)
\(\overrightarrow{u_{AB}}=\left(6;-8\right)\Rightarrow\overrightarrow{n_{AB}}=\left(8;6\right)\)
PTTQ của AB : \(8\left(x-2\right)+6\left(y+7\right)=0\)
\(\Leftrightarrow4x+3y+13=0\)
\(C=CM\cap BC\Rightarrow C\left(5;-6\right)\)
\(\overrightarrow{u_{AC}}=\left(9;-7\right)\Rightarrow\overrightarrow{n_{AC}}=\left(7;9\right)\)
PTTQ của AC : \(7\left(x-5\right)+9\left(y+6\right)=0\)
\(\Leftrightarrow7x+9y+19=0\)
Gọi $A\left( {{x}_{A}};{{y}_{A}} \right);C\left( {{x}_{C}};{{y}_{C}} \right)$
Phương trình đường cao qua $A:\left( d \right):3x+y+11=0$
$\overrightarrow{{{u}_{d}}}=\left( 3;1 \right)\Rightarrow \overrightarrow{AC}.\overrightarrow{u{{ & }_{d}}}=3\left( {{x}_{C}}-{{x}_{A}} \right)+1\left( {{y}_{C}}-{{y}_{A}} \right)=0$
Phương trình trung tuyến qua $C:\left( d' \right):x+2y+7=0$
$d\cap AB=M\left( \dfrac{2+{{x}_{A}}}{2};\dfrac{{{y}_{A}}-7}{2} \right)$
Ta có hệ phương trình: \(\left\{ \begin{array}{l} 3\left( {{x_C} - {x_A}} \right) + {y_C} - {y_A} = 0\\ 3{x_A} + {y_A} + 11 = 0\\ {x_C} + 2{y_C} + 7 = 0\\ \dfrac{{2 + {x_A}}}{2} + 2.\dfrac{{{y_A} - 7}}{2} + 7 = 0 \end{array} \right. \Leftrightarrow \left\{ \begin{array}{l} {x_A} = - 4\\ {y_A} = 1\\ {x_C} = - 1\\ {y_C} = - 8 \end{array} \right.\)
\(\begin{array}{l} \Rightarrow A\left( { - 4;1} \right);C\left( { - 1; - 8} \right) \Rightarrow \overrightarrow {AB} = \left( {2; - 8} \right);\overrightarrow {AC} = \left( {3; - 9} \right);\overrightarrow {BC} = \left( { - 3; - 1} \right)\\ AB:2\left( {x + 4} \right) - 8\left( {y - 1} \right) = 0 \Rightarrow 2x - 8y + 16 = 0\\ AC:3\left( {x + 1} \right) - 9\left( {y + 8} \right) = 0 \Rightarrow 3x - 9y - 69 = 0\\ BC: - 3\left( {x + 1} \right) - 1\left( {y + 8} \right) = 0 \Rightarrow - 3x - y - 11 = 0 \end{array}\)