a) I là điểm trên cạnh BC mà: \[2CI = 3BI \Rightarrow \frac{{BI}}{{CI}} = \frac{2}{3}\]
\( \Rightarrow \frac{{BI}}{{CI + BI}} = \frac{2}{{3 + 2}} \Rightarrow \frac{{BI}}{{BC}} = \frac{2}{5}\)
\( \Rightarrow BI = \frac{2}{5}BC\)
Tương tự ta có: \( \Rightarrow IC = \frac{3}{5}BC\)
J là điểm trên BC kéo dài: \[5JB = 2JC \Rightarrow \frac{{JB}}{{JC}} = \frac{2}{5}\]
\( \Rightarrow \frac{{JB}}{{JC - JB}} = \frac{2}{{5 - 2}} \Rightarrow \frac{{JB}}{{BC}} = \frac{2}{3}\)
\( \Rightarrow JB = \frac{2}{3}BC\)và\(BC = \frac{3}{5}JC\)
Ta có: \(\overrightarrow {AB} = \overrightarrow {AI} + \overrightarrow {IB} = \overrightarrow {AI} - \frac{2}{5}\overrightarrow {BC} = \overrightarrow {AI} - \frac{2}{5}\,.\,\frac{3}{2}\overrightarrow {JB} \)
\( = \overrightarrow {AI} - \,\frac{3}{5}\overrightarrow {JB} = \overrightarrow {AI} - \,\frac{3}{5}\left( {\overrightarrow {JA} + \overrightarrow {AB} } \right) = \overrightarrow {AI} + \,\frac{3}{5}\overrightarrow {AJ} - \,\frac{3}{5}\overrightarrow {AB} \)
\( \Rightarrow \overrightarrow {AB} + \frac{3}{5}\overrightarrow {AB} = \overrightarrow {AI} + \,\frac{3}{5}\overrightarrow {AJ} \)
\( \Rightarrow \overrightarrow {AB} = \frac{5}{8}\overrightarrow {AI} + \,\frac{3}{8}\overrightarrow {AJ} \) (1)
Lại có: \(\overrightarrow {AC} = \overrightarrow {AI} + \overrightarrow {IC} = \overrightarrow {AI} + \frac{3}{5}\overrightarrow {BC} = \overrightarrow {AI} + \frac{3}{5}\,.\,\frac{3}{2}\overrightarrow {JC} \)
\( = \overrightarrow {AI} + \,\frac{9}{{25}}\overrightarrow {JC} = \overrightarrow {AI} + \,\frac{9}{{25}}\left( {\overrightarrow {JA} + \overrightarrow {AB} } \right) = \overrightarrow {AI} - \frac{9}{{25}}\overrightarrow {AJ} + \frac{9}{{25}}\overrightarrow {AB} \)
\( \Rightarrow \overrightarrow {AC} - \frac{9}{{25}}\overrightarrow {AC} = \overrightarrow {AI} - \,\frac{9}{{25}}\overrightarrow {AJ} \)
\( \Rightarrow \overrightarrow {AC} = \frac{{25}}{{16}}\overrightarrow {AI} - \,\frac{9}{{16}}\overrightarrow {AJ} \) (2)
Nhân 2 vế của (1) với \(\frac{5}{2}\) rồi trừ cho vế với vế của (2) ta được:
\(\frac{5}{2}\overrightarrow {AB} - \overrightarrow {AC} = \frac{5}{2}\left( {\frac{5}{8}\overrightarrow {AI} + \,\frac{3}{8}\overrightarrow {AJ} } \right) - \left( {\frac{{25}}{{16}}\overrightarrow {AI} - \,\frac{9}{{16}}\overrightarrow {AJ} } \right)\)
\( = \frac{{25}}{{16}}\overrightarrow {AI} + \frac{{15}}{{16}}\overrightarrow {AJ} - \frac{{25}}{{16}}\overrightarrow {AI} + \frac{9}{{16}}\overrightarrow {AJ} = \frac{3}{2}\overrightarrow {AJ} \)
\( \Rightarrow \overrightarrow {AJ} = \frac{5}{3}\overrightarrow {AB} - \frac{2}{3}\overrightarrow {AC} \)
b) \[\overrightarrow {AG} = \frac{2}{3}\overrightarrow {AH} = \frac{2}{3}\,.\,\frac{1}{2}\left( {\overrightarrow {AB} + \overrightarrow {AC} } \right)\]
\[ = \frac{1}{3}\left( {\frac{5}{8}\overrightarrow {AI} + \,\frac{3}{8}\overrightarrow {AJ} + \frac{{25}}{{16}}\overrightarrow {AI} - \,\frac{9}{{16}}\overrightarrow {AJ} } \right)\]
\[ = \frac{{35}}{{48}}\overrightarrow {AI} - \,\frac{1}{{16}}\overrightarrow {AJ} \]