$\begin{array}{l}
\overrightarrow {AM} = \overrightarrow {AD} + \overrightarrow {DM} = \overrightarrow {AD} + \dfrac{1}{4}\overrightarrow {DC} = \overrightarrow {AD} + \dfrac{1}{4}\overrightarrow {AB} \\
\overrightarrow {DB} = \overrightarrow {DA} + \overrightarrow {DC} = - \overrightarrow {AD} + \overrightarrow {AB} \\
\Rightarrow \overrightarrow {AM} .\overrightarrow {DB} = \left( {\overrightarrow {AD} + \dfrac{1}{4}\overrightarrow {AB} } \right)\left( { - \overrightarrow {AD} + \overrightarrow {AB} } \right)\\
= - A{D^2} - \dfrac{1}{4}\overrightarrow {AB} .\overrightarrow {AD} + \overrightarrow {AD} .\overrightarrow {AB} + \dfrac{1}{4}A{B^2}\\
= - A{D^2} - \dfrac{1}{4}.0 + 0 + \dfrac{1}{4}.4A{D^2}\\
= - A{D^2} + A{D^2} = 0\\
\Rightarrow AM \bot DB
\end{array}$
