# Here’s How Two Teenagers Found a New Proof of the Pythagorean Theorem

## An inspirational example of how elementary math is open to everyone

Recently, two teenage girls — Calcea Johnson and Ne’Kiya Jackson — gave a presentation to the American Mathematical Society where they demonstrated a new proof of the Pythagorean Theorem. Their proof has caused quite a stir.

One reason for the stir delights me and makes me incredibly happy as a teacher of mathematics. By all accounts, these two girls are shaking up the majority male math establishment. This is just an awesome turn of events and one which should inspire anyone — no matter what their background — that excellence in your chosen field of study is always attainable if you have enough joy and passion for what you do.

Another reason for the stir is that the proof these young trailblazers have proposed might make a few established mathematicians eat their words.

This is because their proof uses trigonometry.

Now why is that such a big deal? Well, many of our trigonometric identities and laws depend on the Pythagorean Theorem, and so a number of mathematicians have suggested that any proof of the theorem using trigonometry is circular logic. Put another way, they argue that using trigonometry to prove Pythagoras is basically using A to prove B, when A already depends on B. One strong proponent of this point of view was the mathematician Elisha Loomis, who published a book in 1927 full of non-trigonometric proofs of the theorem, and explicitly stating that trigonometric proofs were impossible.

However, this point of view has been increasingly questioned in recent decades, and a few trigonometric proofs of Pythagoras have made the rounds since then. Claims in the media that Johnson and Jackson’s proof is the first trigonometric proof of Pythagoras are overblown, but their proof could well be the most beautiful and simplest trigonometric proof we have seen to date, and is clearly the work of young, sharp minds uncomplicated by the years of deep research that characterise the work of many experienced mathematicians.

In this article, I’ll do a quick reminder of what the Pythagorean Theorem is, before doing my best to explain how Johnson and Jackson proved it using simple trigonometry. Although their proof hasn’t been published (I hope it will be!), I’ve pieced together their approach from various online…