$\begin{array}{l}
\overrightarrow {KD} = \frac{1}{2}\left( {\overrightarrow {KB} + \overrightarrow {KC} } \right) = \frac{1}{2}\left( {\overrightarrow {KM} + \overrightarrow {MB} + \overrightarrow {KN} + \overrightarrow {NC} } \right)\\
= \frac{1}{2}\left( {\overrightarrow {MB} + \overrightarrow {NC} } \right) = \frac{1}{2}\left( {\frac{1}{2}\overrightarrow {AB} + \frac{1}{2}\overrightarrow {AC} } \right) = \frac{1}{4}\overrightarrow {AB} + \frac{1}{4}\overrightarrow {AC}
\end{array}$