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.
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 :
- is right. This is an application of Thales’ theorem since DG is a diameter.
- is right. This is given.
- . Since , .
- . This is because BC and BG are both radii of and is isosceles.
- . 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:
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: