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Pythagoras' theorem has a variety of geometric proofs, such as:

Proof of the pythagoras' theorem

I want to teach at least one of these proofs to my high school students, because it shows that the formula $\|(x,y)\| = \sqrt{x^2 + y^2}$ doesn't just come out of nowhere.

However I'm a bit confused about what these "proofs" are actually achieving.

From a higher mathematics perspective, Pythagoras' theorem isn't a theorem, it's a definition. We define the distance between two points in $\mathcal{l}^2(\{e_1,e_2\})$ a certain way, because that's how the space $l^p(X)$ is defined. Or, alternatively, we can think of $l^2(X)$ as an inner product space, and recover the metric by defining $$\|x\| = \sqrt{\langle x, x\rangle}.$$ In either case, these kinds of geometric proofs play no role, and the distance between two points is either by definition, or is recovered almost immediately from the definitions.

In light of this, I'm not sure what these geometric arguments are really telling us. I think what's going on is that if we assume $\mathbb{R}^2$ is a measure space and a metric space in a compatible way, then these geometric proofs show us that the metric structure has to be a very specific structure. However, I don't understand the details. So, my question is:

Question. What is the epistemological status of the usual proof(s) of Pythagoras' theorem?

goblin GONE
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    I think you are mixing up axiom systems. If you start with Euclid's axioms of plane geometry, then Pythagoras' theorem is indeed a theorem. If you start with coordinate geometry in R^2, then Pythagoras' theorem is the definition of a metric, and then you can prove Euclid's axioms. –  Oct 10 '19 at 05:18
  • @Rahul, I didn't realize Euclid's axioms of plane geometry actually had a notion of distance. Is the idea (roughly) to identify the elements of $\mathbb{R}_{>0}$ with equivalence classes of line segments? If not, where does distance come from in the Euclidean approach? – goblin GONE Oct 10 '19 at 05:30
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    @goblin Yes, that's the idea. However, if you really want to do it rigourously, Euclid's axioms are hopelessly incomplete and inadequate. I suggest you take a more modern axiom system, like Hilbert's. – Arthur Oct 10 '19 at 05:41
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    Euclid doesn't have the real numbers, he has congruence and rearrangement. Distances (resp. areas) are measured as ratios relative to an arbitrarily chosen unit line segment (resp. square with unit length sides). For example, if two copies of a given line segment placed end to end are congruent to three copies of the unit line segment, then the length of the former line segment is said to be $3/2$. Pythagoras' theorem is a statement about the areas of the squares constructed on the three sides of a right triangle. –  Oct 10 '19 at 05:43
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    Look up "synthetic geometry" for more about this sort of thing. –  Oct 10 '19 at 05:45
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    One of the most rigorous textbook treatments of planar geometry is in Hartshorne's book. An alternative to Euclid/Hilbert axioms (which indeed do not have a notion of a distance at least at first) is Birkhoff's axioms, treated for instance in Moise's book. With these axioms, the notion of a distance appears already in the axioms. See also https://arxiv.org/pdf/1302.1630.pdf – Moishe Kohan Oct 16 '19 at 02:08
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    As for proofs of Pythagorean Theorem using area, there is a huge underlying issue you should be aware of, see my answer here: https://math.stackexchange.com/questions/675522/whats-the-intuition-behind-pythagoras-theorem/676731#676731 – Moishe Kohan Oct 16 '19 at 02:15
  • @MoisheKohan, thanks! That's very helpful. – goblin GONE Oct 17 '19 at 02:59
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    This question discusses Pythagoras' theorem in more well founded systems than Euclidean geometry and should be of interest. – Jam Oct 29 '19 at 01:39

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