The Simplest Proof of the Pythagorean Theorem

Simple and Amazing

It is common to find demonstrations of the Pythagorean theorem through the calculation of areas, especially because its formula is made up of quadratic powers. But we developed a more primordial methodology as it does not depend on calculating areas, just analyzing angles. For this we will use a geometric construction based on a circle.

Firstly, be the right triangle ABC, where the 90º angle is at the vertex A. The hypotenuse measures “a” and the legs “b” and “c”, as shown in the figure.

Let’s draw a circle with center at “C” and radius “c”, that is, this circle will contain the vertex “A”

Let’s now draw a straight line from “B” passing through “C”,

This will determine the points “N” and “M” (diametrically opposite).

We now have two new auxiliary triangles: “ABN” and “MBA”,

both are similar to each other, that is, they maintain a proportion between the dimensions of their corresponding sides. This is because they have two commons angles, that is: the angle with the vertex in “B” ​​and the angle β:

According to the following deduction:

since,

then,

and, since triangle “NCA” is isosceles, then:

Once the geometric similarity has been proven, we can write the proportional relationship between the corresponding sides:

And, using the corresponding literal values:

or

and finally