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I recently came across a proof of the Pythagorean Theorem that was new to me that gave me an aha! moment. This was given in Sanjay Gulati's excellent "Mathematics Academy" blog as a Geogebra demonstration. He does not indicate the original source of the proof. The aha! moment comes for the connection between the Pythagorean Theorem and an apparently unrelated theorem that I always teach in my elementary geometry class, the "crossed chords" theorem. The aha! moment occurs from looking at the following picture.

Then the crossed-chords theorem tells us that (c + a)(c - a) = b

^{2}, or c

^{2}- a

^{2}= b

^{2}.

## 2 comments:

I don't necessarily see anything "wrong" with it. But it is definitely a less intuitive way of coming to same conclusion, so probably not the simplest way to teach a class to prove the Pythagorean theorem. But then again, perhaps the fact that it is not as straight forward and incorporates several other things might be a good thing, at least for certain audiences. It reminds them about FOIL and the crossed-chords theorem, as well has helping them to see that there are multiple ways to look at and solve every problem, even in math, which I think a lot of math and physics teachers don't emphasize quite as much as they should! You never know what students that method might make more sense to than the more common methods also.

The proof looks valid to me too. Is there a really well hidden circular argument? My proof of the "crossed-chords" theorem (or "butterfly theorem" as I think of it) doesn't use Pythag, and the diagram is valid for any right triangle, so it seems everything is fine.

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