If the length of vector ${\bf a}$ is $a=|{\bf a}|$, then we have $a^2={\bf a}\cdot {\bf a}$.

We have $\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}={\bf b} - {\bf a}$.

It might be easier to consider $(AB^2+CD^2) - (AC^2+BD^2)$ and show that this is equal to $0$ rather than trying to show that $AB^2+CD^2 = AC^2+BD^2$.