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 Prove that the sum of the squares of ...
 Entrance Exam
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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 d_{1} and d_{2} 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 + x_{0}, y) and D have the coordinates (x_{0}, y). If x_{0} is 0 then the parallelogram is a rectangle.
Firstly, using the distance formula, let's solve for the length of the two diagonals, d_{1} and d_{2}.
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.