trong mp oxy cho ΔABC vs A=(5,4), B=(2,7), C=(-2,-1) a. Tìm trọng tâm G, trực tâm H và tâm I đường tròn ngoại tiếp ΔABC b.CM I,G,H thẳng hàng

Đáp án:

$\begin{array}{l}
a)\\
\left\{ \begin{array}{l}
{x_G} = \frac{{{x_A} + {x_B} + {x_C}}}{3} = \frac{5}{3}\\
{y_G} = \frac{{{y_A} + {y_B} + {y_C}}}{3} = \frac{{10}}{3}
\end{array} \right. \Rightarrow G\left( {\frac{5}{3};\frac{{10}}{3}} \right)
\end{array}$

$\begin{array}{l}
H\left( {x;y} \right) \Rightarrow \left\{ \begin{array}{l}
\overrightarrow {AH}  = \left( {x – 5;y – 4} \right)\\
\overrightarrow {BH}  = \left( {x – 2;y – 7} \right)\\
\overrightarrow {BC}  = \left( { – 4; – 8} \right)\\
\overrightarrow {AC}  = \left( { – 7; – 5} \right)
\end{array} \right.\\
Do:\left\{ \begin{array}{l}
AH \bot BC\\
BH \bot AC
\end{array} \right. \Rightarrow \left\{ \begin{array}{l}
\overrightarrow {AH} .\overrightarrow {BC}  = 0\\
\overrightarrow {BH} .\overrightarrow {AC}  = 0
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
 – 4\left( {x – 5} \right) – 8\left( {y – 4} \right) = 0\\
 – 7\left( {x – 2} \right) – 5\left( {y – 7} \right) = 0
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
x = \frac{{11}}{3}\\
y = \frac{{14}}{3}
\end{array} \right. \Rightarrow H\left( {\frac{{11}}{3};\frac{{14}}{3}} \right)\\
I\left( {x;y} \right)\\
 \Rightarrow \left\{ \begin{array}{l}
A{I^2} = B{I^2}\\
A{I^2} = C{I^2}
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
{\left( {x – 5} \right)^2} + {\left( {y – 4} \right)^2} = {\left( {x – 2} \right)^2} + {\left( {y – 7} \right)^2}\\
{\left( {x – 5} \right)^2} + {\left( {y – 4} \right)^2} = {\left( {x + 2} \right)^2} + {\left( {y + 1} \right)^2}
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
x = \frac{2}{3}\\
y = \frac{8}{3}
\end{array} \right. \Rightarrow I\left( {\frac{2}{3};\frac{8}{3}} \right)
\end{array}$

b)

$\begin{array}{l}
\overrightarrow {IH}  = \left( {3;2} \right);\overrightarrow {IG}  = \left( {1;\frac{2}{3}} \right)\\
 \Rightarrow \overrightarrow {IH}  = \frac{3}{2}.\overrightarrow {IG} 
\end{array}$

Vậy I,H,K thẳng hàng