Q:

An intuitive proof for one of the  fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the
  squares of the four sides of a parallelogram."

I find this property very useful while solving different problems on Quadrilaterals & Polygon,so I am very inquisitive about a intuitive proof of this property.

A:

Let two sides of the parallelogram correspond to the vectors $\vec a$ and $\vec b$. Then the diagonals correspond to the vectors $\vec a+\vec b$ and $\vec a-\vec b$, and
$$(\vec a+\vec b)^2+(\vec a-\vec b)^2=2\vec a^2+2\vec b^2\;,$$
which is the desired identity between the sums of the squares of the diagonals and the sides.

