$A ( 1 ; -3 ) , B ( -1 ; 5 ) , C ( 3 ; - 2 )\\ a)\overrightarrow{AB}=(-2;8);\overrightarrow{AC}=(2;1)\\ b)G(1;0);I\left(1;\dfrac{3}{2}\right)\\ c)H(x;y)\\ \overrightarrow{CH}=\left(x-3;y+2\right);\overrightarrow{BH}=\left(x+1;y-5\right)$
$H$ là trực tâm $\Delta ABC$
$\Rightarrow\left\{{}\begin{matrix}CH\perp AB\\BH\perp AC\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\overrightarrow{CH}.\overrightarrow{AB}=0\\\overrightarrow{BH}.\overrightarrow{AC}=0\end{matrix}\right.\\ \Leftrightarrow \left\{\begin{array}{l} -2(x-3)+8(y+2)=0\\ 2(x+1)+y-5=0\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} -2x+8y=-22\\ 2x+y=3\end{array} \right.\\ \Leftrightarrow \left\{\begin{array}{l} x=\dfrac{23}{9}\\y=-\dfrac{19}{9}\end{array} \right.\\ \Rightarrow H\left(\dfrac{23}{9};-\dfrac{19}{9}\right)$
