Prove that the sum of the squares of the diagonals of parallelogram is equal to the sum of the squares of its sides.

In ABCD, let d₁ and d₂ be the diagonals AC and BD, respectively. Construct a coordinate system so that A is the origin, B lies on the positive x axis, and C and D lies above the x axis. See the diagram below. Assume that AB is equal to DC and AD is equal to BC. We must prove that .

Let x be the x coordinate of B, let C have coordinates (x + x₀, y) and D have the coordinates (x₀, y). If x₀ is 0 then the parallelogram is a rectangle.

 

Firstly, using the distance formula, let's solve for the length of the two diagonals, d₁ and d₂.

Now let's sum the squares of the diagonals:

Secondly, again using the distance formula, let's solve for the length of the two sides AB and AD.

Finally, we want to sum the squares of the sides and multiply by 2 (4 sides total, 2 of each length).

Thus, we have proven that the sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the diagonals.