Feedback | SoME

The SIMPLEST Pythagorean Theorem Proof

Audience:

Tags: geometryproof

This video is a (to the best of my knowledge) new proof of the Pythagorean Theorem which I came up with. It involves slicing two circles and rearranging the pieces to make a larger circle. Then, the viewer is challenged to generalize this proof to prove the Law of Cosines and this proof is drawn. I think this video is perfect for a high school geometry class, as a video to be played for the class after they learn about the Pythagorean Theorem (or the Law of Cosines). I hope that it will offer a more visually appealing proof than the ones students would otherwise encounter (and one with infinite rotational symmetries!) I think that the challenge of generalizing the proof to non-right triangles is also a great one for teachers to pose to students in the classroom.

Analytics

5 Overall score*

120 Rank

17 Votes

13 Comments

Comments

Better than most. The title could use some work as it is the only thing that is the hook for the piece. But using circles to show this concept was great

I love the beautiful visuals and the creative solution. One suggestion I have is to show what it would look like for the triangle pieces to get infinitely thin and fully become the circle. That way, students unfamiliar with that kind of proof can understand.

4.1

I really liked the visualizations and the smooth movements of the shapes from circle to circle. The generalization from the Pythagorean Theorem to the Law of Cosines was also implied quite clearly. The main problem I have is that it’s not very rigorous and doesn’t have any further resources noted anywhere (description or at the end).

5.3

It’s elegant but I do miss a spoken explanation of it on top of this video. And since it is so short it would be nice to dive in a bit to see why this method works and how I could apply it in other situations

Nice.

3.9

Neat, but lacked the rigor that I would expect from something calling itself a “proof”. It’s a very cool visual demonstration, and would make a nice companion to a longer-form video that goes into more detail.

These are cool visual proofs and could be a fun exercise to do in class. Here’s my feedback. The Pythagorean Theorem proof is not super easy to follow. The way the green circle is laid around the blue one isn’t obvious without extra explanation. Further, a student probably needs to understand calculus concepts for this proof to click. I also think an unsuspecting viewer could get the impression that a circle of radius 5 and an annulus with thickness 5 have the same area. In the law of cosines proof, the outer annulus deconstruction into a circle and a parallelogram is a pretty steep ask for geometry students. Among other things, I’m not sure you’d be able to convince them the length of the parallelogram is equal to the circumference of the small circle.

6.8

Cool original proof, especially for how it generalizes (calling it “simplest” isn’t really the idea though, I think the hook is it being an original circle dissection and generalizing to Law of Cosines). It would be good to show how this also implies putting semicircles on the sides of the a,b,c right triangle has the semicircle areas equal (this is a picture I’ve seen before, but never as a proof of Pythag more a takeaway once you understand similarity / scaling arguments).

More time and depth would have been good to show for the details of what the limiting process really is. The proof without words went pretty fast, and for me to really understand the argument I would have needed to see a closer look into the details of the shapes, especially in how it generalized to Law of cosines. What the missing wedges are that equalled the big parallelogram wasn’t clear, and I don’t think it could have been clarified in just 2 minutes without words.

I really liked the visualisation of where the cosine law came from. It’s animated nicely, and the video isn’t too long.

Not clear enough to be a proof without words.

6.3

A very interesting proof that I haven’t seen before (or at least don’t remember)! Kudos if you really are the first to discover it. (though I wouldn’t say it’s the simplest.)

I could follow the initial argument very well, though you kind of lost me on the generalization at the end, where some more explanation and/or slowness would have helped.

I suspect you’ll get a lot of comments saying it lacks rigor and demanding limits etc., but for a quick aha over lunch break, I don’t mind much.

The proof reminded me of Mamikon Mnatsakanian’s “visual calculus”, which (on the off chance you’re not already familiar) is worth checking out!

4.5

Maybe try to add step by Step explanation, why the area has to to something with the cosine. Also maybe try to add something catching, interesting or maybe even funny to make it really memorable

3.8

Really interesting method. I liked the animations but felt the pacing is too fast and unclear in parts. It’s also not very clear why we should learn this approach, considering the Pythagorean theorem has so many other proofs as well.