High school students who came up with 'impossible' proof of Pythagorean theorem discover 9 more solutions to the problem

By proving Pythagoras' theorem using trigonometry, but without using the theorem itself, the two young women overcame a failure of logic known as circular reasoning.

Before these young ladies came up with these proofs, the only way people could come up with to balance the equation was to use things that boiled down to the actual thing they were trying to prove. It's like saying all things are made of atoms, but then people say well what the heck are atoms made of, smartypants?! So these girls found the Higgs-Boson of trigonometry while in high school as a piece of extra credit on a test. That's my understanding as a low B, high C math student.

Edit: For any of you that are mathy, here is their actual paper.

From the Conclusion of the paper:

The reader may be surprised to learn that the catalyst for us to start this project was a bonus question of a high school math contest. The bonus question was to create a new proof of the Pythagorean theorem. Motivated by the $500 prize, we independently decided to take on this task. It proved to be much harder than we first imagined, and we each spent many long nights trying and failing to create a proof. After roughly a month of mental labor, we each completed and submitted our work. Mr. Rich, a math volunteer at our high school, believed our proofs were novel enough to be presented at a mathematical conference. Neither of us had such confidence in our work at that point, but we decided to go along with it anyway. This is when we began to work together.

For the next two to three months, we spent all of our free time perfecting and polishing our work. We worked both independently and together after school, on weekends, and even during holidays. In the process, with Mr. Rich as our faculty advisor, we created additional proofs. We did all of this not knowing if we would even be allowed to present at the conference, which is usually only done by professional mathematicians, and occasionally college students. To our surprise, our high school work was taken seriously, and we were approved to present at the American Mathematical Society’s Southeastern Sectional conference in March of 2023. Being the youngest people in the room and the youngest presenters was terrifying, but knowing that this was the culmination of all of our previous efforts gave us the confidence to present.

We were then encouraged by the AMS to submit our findings to an academic journal. This proved to be the most daunting task of all, since we had absolutely no experience writing for an academic journal. We were both also dealing with the stressors that come with adjusting to the college environment. Learning how to code in LaTeX is not so simple when you’re also trying to write a 5 page essay with a group, and submit a data analysis for a lab. With the guidance and wisdom of our mentors, and a lot of personal dedication, we were able to craft this paper. The support of our family and later our community helped us to persevere. Our journey to this point was by no means simple or straightforward. There was no road map laid out for us, and there certainly was no guarantee that any of our work would go further than our own heads. There were many times when both of us wanted to abandon this project, but we decided to persevere to finish what we started.

It’ll sound like splitting hairs, but I’ll try:

Trigonometry is based on the Pythagorean theory being true. They proved the Pythagorean theorem effectively in reverse without using the theorem itself as a basis. So they used the structure of trigonometry to prove the basic underlying principle of trigonometry. Bad analogy: kind of like if you have an airplane first, and THEN you worked out the physics of lift. You knew it could fly and how to fly it, but never questioned how it worked.

I'm not that great either but to my understanding you are right. The thing is by giving a solid proof foundation to what was mostly glued together by basic understanding we can now build over it and arrive to new things.