### Analytic Proofs

Analytic proofs of geometric theorems can often be demonstrated more clearly by the use of coordinates as opposed to synthetic proofs which use deduction from axioms and previously derived theorems.

### Problem 1. The Parallelogram Law

Prove analytically that the sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of the

diagonals.

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.

super