$\begin{array}{l}
\overrightarrow {AM} = \dfrac{1}{3}\overrightarrow {AB} ,\overrightarrow {BN} = \dfrac{1}{3}\overrightarrow {BC} ,\overrightarrow {CP} = \dfrac{1}{3}\overrightarrow {CA} \\
\Rightarrow \overrightarrow {AN} + \overrightarrow {BP} + \overrightarrow {CM} = \overrightarrow {AB} + \overrightarrow {BN} + \overrightarrow {BC} + \overrightarrow {CP} + \overrightarrow {CA} + \overrightarrow {AM} \\
= \left( {\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CA} } \right) + \left( {\overrightarrow {BN} + \overrightarrow {CP} + \overrightarrow {AM} } \right)\\
= \overrightarrow 0 + \left( {\dfrac{1}{3}\overrightarrow {BC} + \dfrac{1}{3}\overrightarrow {CA} + \dfrac{1}{3}\overrightarrow {AB} } \right)\\
= \overrightarrow 0 + \dfrac{1}{3}\left( {\overrightarrow {BC} + \overrightarrow {CA} + \overrightarrow {AB} } \right)\\
= \dfrac{1}{3}.\overrightarrow 0 = \overrightarrow 0
\end{array}$
