The two-circles method of proving the Pythagorean Theorem

The Pythagorean theorem, stating that the square of the hypotenuse is equal to the sum of the squares of the sides in a right triangle, is a fundamental concept in geometry and trigonometry.

There are many, many ways to prove the Pythagorean theorem. This page contains 84 different proofs, and The Pythagorean Proposition by Loomis contains another 367 proofs.

This is what I think to be an interesting proof, by the use of two circles. It is number 89 in Loomis’s book.

Starting with a right triangle, we draw two circles:

Here, is right, and A and B are the centers of circles and respectively.

Now we draw some additional lines, extending AB to F and G:

In this diagram, we can prove that :

1. is right. This is an application of Thales’ theorem since DG is a diameter.
2. is right. This is given.
3. . Since , .
4. . This is because BC and BG are both radii of and is isosceles.
5. .
6. . The two triangles have two shared angles: and .

In a similar way, we can prove that .

The rest of the proof is algebraic rather than geometric. Let’s call the side AC to be b, BC=a, and AB=c.

From the similar triangles, we have the following ratios:

(or, )

(or, )

Adding the two equations, we get:

The line BF can be split into AF and AB which is equal to c+b since AF = AC.

The line EB can be considered the difference between AB and AE, which is equal to c-b.

Similarly, AG = AB+BG = c+a, and AD = AB-DB = c-a. By substitution:

Q.E.D.