Pythagorean´s theorem is one of the most important theorems in the broad field of mathematics. It is important for norms, trigonometry and so on.

Let´s briefly discuss the theorem. In a perpendicular triangle we have

Each variable represents one of the three sides and c is the hypotenuse, the longest side in a perpendicular triangle. The area of a square is defined as the multiplication of it´s sides. So we actually look at a statement about areas of squares. The Pythagorean theorem is a nice and easy to understand example of different mathematical objects that are connected: squares and perpendicular triangles.

 

The idea to prove the theorem is as follows:

We add our triangle to each side of c² so that we get a bigger square. We will get this:

Now, we will look at two different ways to compute the area of this big square and we will be able to proof the Pythagorean theorem.

  1.  We know from elementary geometry that the area of each triangle is half of the area of a corresponding rectangle. So the area of all triangles   in our big square is

Now we simply have to add c² to this to get the entire area

2. We know that each side of our big square is a+b. So, we have (a+b)² as area. Now, we can use one of the binomial theorems to find another             formula to describe the area of our square

Since both formulas describe the same thing it means they are equal and we can create the equation that will lead to the Pythagorean theorem. Simply substract 2ab from both sides

 

Sources:

David Acheson – The calculus story, p.9

Pictures from wikipedia

Latex output from https://www.codecogs.com/latex/eqneditor.php

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