Geometric interpretation of $\|a+b\|^2 +\|a-b\|^2 = 2 \|a\|^2 +2\|b\|^2$ For vectors $a,b$ in an inner product space, I have just proven the identity:
$$\|a + b\|^2 + \|a - b \|^2 = 2\|a\|^2 + 2\|b\|^2$$
Now I am trying to visualise this in $\mathbb{R}^2$, but I'm having a hard time drawing out what this means. Isn't 
it basically saying that, in a parallelogram the squared lengths of the diagonals add up to the squared lengths of 
the sides of the parallelogram?
We can see that $A+B$ and $A-B$ represents the diagonals while vectors $A$ and $B$ are the sides of the parallogram.
No. It means that the sum of the squares of the diagonals is equal to the sum of the squares of all sides.