There are lots of geometric dissection and reassembling proofs of the Pythagorean Theorem (PT). I want to know what are the necessary ideas to allow a proof using discrete instances.

For example, we can, not too terribly painstakingly, using elementary tiling of a diagram show that a 1,1 right triangle has area 2 square on the hypotenuse

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and even a 3,4 right triangle has 5 as its hypotenuse all without using PT.

Could these instances (or others) be used prove the full generality of PT, assuming some ideas of continuity, some light inference on the relations of the three values?

The idea is simply one of analogy: two polynomials of degree $d$ are equal if they coincide on $d+1$ points. Here, instead of 2 variables ($x$ and $y$) we have 3 ($a,b,c$).

So for a (very loosely analogous) proof of PT, what would be needed?

  • what assumption of form of the relation? $d a^2 + e b^2 = f c^2$ seems like too much to assume, practically assuming PT! (but if that's what it takes...)
  • how many instances? (obviously 3 for the three vars $d,e,f$, but can you do it in fewer?
  • any continuity axioms? (or minor handwaving justifications)
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    $\begingroup$ There are some differences between these two situations. For polynomial interpolation (say, using Lagrange's formula), you already know that the function you're looking for is a polynomial. This is huge, as most functions are not polynomials. In the case of a right triangle, where you're trying to figure out the relationship between the lengths of the three sides (this is half of the Pythagorean Theorem; there's also the converse), you don't know a priori that the relationship is polynomial. $\endgroup$ – Sammy Black Nov 6 '15 at 22:40
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    $\begingroup$ @SammyBlack Understood. It may very well be that to establish the form of the relation among the sides that you've already gotten the constants and proven PT. And if you knew the form were $d a^2 + e b^2 = f c^2$ only one instance would set the coeffs to 1. I realize my "any other assumptions" is a bit broad, but are there any restrictions on form that leave a non-trivial interpolation? $\endgroup$ – Mitch Nov 7 '15 at 2:42
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    $\begingroup$ Also, would take 3 instances if the form were as given in my comment and edited question. Is it that simple? Also, is there any quick intuition that would make someone guess that the relation is some combination of areas? $\endgroup$ – Mitch Nov 10 '15 at 1:17
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    $\begingroup$ @Mitch I don't know if this is what you want, but if you accept Heron's formula as true, PT'll be a consequence, i.e., let S = area of any triangle where the three sides are a, b, and c, and let s = 1/2 (a+b+c), Heron's formula says S = sqrt(s(s-a)(s-b)(s-c)); if we assign a and b such that a is the base and b is the height of the triangle, it follows that if our triangle is a right triangle, S = 1/2 (a.b). Solving 1/2 (a.b) = sqrt(s(s-a)(s-b)(s-c)), we get c^2 = a^2 + b^2. $\endgroup$ – Damkerng T. Nov 15 '15 at 11:38
  • $\begingroup$ The Pythagorean Theorem can be proven using nothing more than the properties of similar right triangles. There are also many proofs using areas of rectangles and some properties of right triangles. I am not sure what you are asking. Can you clarify? The places you claim do not use PT are essentially reproving PT (some proofs of PT are very simple). $\endgroup$ – robjohn Nov 15 '15 at 17:07

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