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Today is a rare day! The date that this article is being posted is 10/24/26. Which, just so happens to be a Pythagorean triple! I.e., $10^2+24^2 = 26^2.$ If it were not for the fact that there is already an article on my site about Pythagorean triples, today would have been a perfect day to write one. But, we don’t need to articles about Pythagorean triples on this cite, so today will be an article following one proof of the Pythagorean theorem. It’s written in a format to give you the proof slowly, so if you want to try to prove it on your own you can try!

The Pythagorean Theorem – A Beauty

The Pythagorean Theorem isn’t only one of the most useful theorems in mathematics, it’s also one of the most beautiful! Whenever mathematicians get together to discuss the most beautiful results in math, the Pythagorean theorem almost always is in the discussion. Here’s a wonderful factoid, there are more than 350 different proofs out there for you to enjoy! Today, I wanted to cover one of the most famous, and I wanted to do it with minimal text. Let’s get going!

The Pythagorean Theorem: Let a right triangle’s two legs have length $a$ and $b$ as well as a hypotenuse with length $c.$ Then,

$$a^2+b^2=c^2.$$

Proof: (Click to see proof)

Let’s draw a the right triangle described as well as implementing the trick where we interpret $a^2, b^2,$ and $c^2$ as areas of squares! Then, we draw those squares so they align with the triangle’s sides,

Thus, our new goal is to prove that the red area plus the blue area equal the purple area. I don’t want to give away the answer just yet, so take a moment to think of a way you can rearrange the squares to make a larger square. Hint: you will need to use four triangles.

Click to see more of the proof!

Here you go!

Note that we know that these puzzles made of squares and triangles are themselves squares, with side lengths $a+b.$ This is because the right triangles have angles that sum to a straight angle, that is, $\angle ac + \angle bc + 90^{\circ} = 180^{\circ}.$

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Short and Sweet

I would like to make one final comment, more of a disclosure. The proof above isn’t a full-on Proof. We didn’t justify every step using the Axioms of geometry, or quote theorems that are proved using those axioms. For example, how did we know that we could construct squares with side lengths $a,b,$ and $c?$ Any who, I’m sure (hope) you are comfortable assuming the things that we did, and this is why this proof was good enough for us—or at least for me! It gives the spirit of the proof and the joy of how creative the proof is. With that said, you can check out Euclid’s fully justified and rigorous proof found in his book The Elements. It’s beautiful how carful, creative, and rigorous he is. But, for those who don’t want to read it, check out these amazing lecture series,

• Alex Kontorovich’s lectures on the History of Mathematics
• Alex Kontorovich’s lectures on Euclidean Geometry