Áp dụng quy tắc trung điểm, ta được:
$\overrightarrow{AK} = \dfrac{1}{2}(\overrightarrow{AM} + \overrightarrow{AN})$
$\overrightarrow{AD} = \dfrac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$
Ta có:
$\overrightarrow{KD} = \overrightarrow{KA} + \overrightarrow{AD}$
$=-\dfrac{1}{2}(\overrightarrow{AM} + \overrightarrow{AN}) + \dfrac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$
$= - \dfrac{1}{2}\left(\dfrac{1}{2}\overrightarrow{AB} + \dfrac{1}{3}\overrightarrow{AC}\right) + \dfrac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC})$
$ = \dfrac{1}{2}\overrightarrow{AB} - \dfrac{1}{4}\overrightarrow{AB} + \dfrac{1}{2}\overrightarrow{AC} - \dfrac{1}{6}\overrightarrow{AC}$
$=\dfrac{1}{4}\overrightarrow{AB} + \dfrac{1}{3}\overrightarrow{AC}$