\[\begin{array}{l}
\overrightarrow {BC} = \overrightarrow {AC} - \overrightarrow {AB} = - \overrightarrow {AB} + \overrightarrow {AC} \\
\overrightarrow {AM} = \dfrac{1}{2}\left( {\overrightarrow {AD} + \overrightarrow {AE} } \right)\\
\Rightarrow \overrightarrow {BC} .\overrightarrow {AM} = \left( { - \overrightarrow {AB} + \overrightarrow {AC} } \right).\dfrac{1}{2}\left( {\overrightarrow {AD} + \overrightarrow {AE} } \right)\\
= \dfrac{1}{2}\left( { - \overrightarrow {AB} .\overrightarrow {AD} + \overrightarrow {AC} .\overrightarrow {AD} - \overrightarrow {AB} .\overrightarrow {AE} + \overrightarrow {AC} .\overrightarrow {AE} } \right)\\
= \dfrac{1}{2}\left( { - \overrightarrow {AB} .\overrightarrow {AD} + 0 - 0 + \overrightarrow {AC} .\overrightarrow {AE} } \right)\\
= \dfrac{1}{2}\left( {AB.AD - AC.AE} \right) = 0(do\,AB = AE,AC = AD)\\
\Rightarrow AM \bot BC
\end{array}\]
