Geometric interpretation of $\|a+b\|^2 +\|a-b\|^2 = 2 \|a\|^2 +2\|b\|^2$

Question

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?

Answer 1

We can see that $A+B$ and $A-B$ represents the diagonals while vectors $A$ and $B$ are the sides of the parallogram.

Answer 2

No. It means that the sum of the squares of the diagonals is equal to the sum of the squares of all sides.