What is the geometric interpretation of $\|a-b\|^2+\|a+b\|^2=2\|a\|^2 + 2\|b\|^2$? I thought something with a triangle, but I couldn't picture that out with drawing the vectors.
I have the answer below, but just noticed the homework tag. Thinking of a triangle is close..
If you just want hints: What four sided figure would $a$ and $b$ determine? And, how do the vectors $a+b$ and $a-b$ relate to this figure?
The sum of the squares of the lengths of the diagonals of a parallelogram is equal the sum of the squares of the lengths of the sides of the parollelogram.