F là trung điểm AB $\Rightarrow \overrightarrow{\text{AF}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AB}}$; E là trung điểm AC $\Rightarrow \overrightarrow{\text{AE}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AC}}$

Ta có EF song song BC (đường trung bình)

Mà D là trung điểm BC $\Rightarrow$ I là trung điểm EF $\Rightarrow$ AI là trung tuyến $\Delta AEF$

$\Rightarrow \overrightarrow{\text{AI}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AE}}\text{ + }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AF}}$

Theo tính chất trọng tâm:

$\overrightarrow{\text{AG}}\text{ = }\frac{\text{2}}{\text{3}}\overrightarrow{\text{AD}}\text{ = }\frac{\text{2}}{\text{3}}\left(\frac{\text{1}}{\text{2}}\overrightarrow{\text{AB}}\text{ + }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AC}}\right)\text{ = }\frac{\text{2}}{\text{3}}\left(\overrightarrow{\text{AE}}\text{ + }\overrightarrow{\text{AF}}\right)\text{ = }\frac{\text{2}}{\text{3}}\overrightarrow{\text{AE}}\text{ + }\frac{\text{2}}{\text{3}}\overrightarrow{\text{AF}}$

DE là đường trung bình tam giác ABC

$\Rightarrow \overrightarrow{\text{DE}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{BA}}\text{ = -}\frac{\text{1}}{\text{2}}\overrightarrow{\text{AB}}\text{ = -}\overrightarrow{\text{AE}}$  hay $\overrightarrow{\text{DE}}\text{ = -}\overrightarrow{\text{AE}}\text{ + 0 . }\overrightarrow{\text{AF}}$

D là trung điểm BC $\Rightarrow \overrightarrow{\text{DC}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{BC}}$

$\Rightarrow \overrightarrow{\text{DC}}\text{ = }\frac{\text{1}}{\text{2}}\overrightarrow{\text{BA}}\text{ + }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AC}}\text{ = -}\frac{\text{1}}{\text{2}}\overrightarrow{\text{AB}}\text{ + }\frac{\text{1}}{\text{2}}\overrightarrow{\text{AC}}\text{ = -}\overrightarrow{\text{AE}}\text{ + }\overrightarrow{\text{AF}}$