Help with complex numbers

Show that $\displaystyle

\left| {a + b} \right|^2 + \left| {a - b} \right|^2 = 2\left( {\left| a \right|^2 + \left| b \right|^2 } \right)

$

where a and b are complex numbers. Interpret this results geometrically.

$\displaystyle

\begin{array}{l}

\left| {a + b} \right|^2 + \left| {a - b} \right|^2 \\

= \left| a \right|^2 + 2\left| a \right|\left| b \right| + \left| b \right|^2 + \left| a \right|^2 - 2\left| a \right|\left| b \right| + \left| b \right|^2 \\

= 2\left| a \right|^2 + 2\left| b \right|^2 \\

= RHS \\

\end{array}

$

I don't know how to interpret this result geometrically though, any help would be appreciated.