Hi Dave,

> Do you have a good or favorite proof or way to see it intuitively???

There's a multitude of proofs of Pythagoras' theorem! Even
President Garfield got into the act and provided a proof in 1876.

To answer your question, I do have a favorite, and it does help
me "see" the result. It uses several ideas that I like.
There's the idea seen in Combinatorics of establishing an
identity by counting the same thing in two different ways.
There's also the idea seen in Probability of calculating a
probability indirectly by directly calculating the probability
of everything else that could happen. And then there's the
seemingly unhelpful idea of adding clutter to the mix by
working with several copies of the triangle instead of just
the one copy in the statement of the theorem. To me, that last
idea is the creative leap. Everything else is just details,
in comparison.

I think that's enough talk about ideas used in the proof.
Here's the proof:

The basic plan is to look at the area left over when four
copies of the triangle are used to cover up some of the area
of a large square. One way of arranging the four triangles
results in the uncovered area being h^2, while another way
results in the uncovered area being a^2+b^2.

Here are the details:

For the first arrangement, take four copies of the a,b,h
triangle and attach them via the hypotenuse to each side of
a square with side length h. This results in a larger square
of side length a+b, with a "hole" of size h^2.

For the second arrangement, take the four triangles and combine
pairs of triangles to form two rectangles with sides of length
a and b. Now take these two rectangles and arrange them so
that one is rotated 90 degrees relative to the other, and
they share a vertex. They're inside a square with side length
a+b, with two "holes", one of size a^2, the other of size b^2.

As for the generalization to general triangles:
I don't have a way to intuitively understand it, other than
as a perturbation of the right angle version of the theorem.
If you come across an approach that sheds some light on
the result, please let me know.

It's great to hear from you Dave! It's been a long time since
the SB Aquatic team got together! I'd almost forgotten it once
existed.

All the best,

Walter

--- monkeyEinstein wrote:
> Hi Walter,
> This is Dave C. of the StonyBrook Aquatic team.
> I was looking at you connections on Pythagoras.
> Although I know how to "prove" a^2 + b^2= h^2 for a,b,h=hypoteneuse
> the sides of a rt. triangle, I still have a hard time seeing why in
> an intuitive way.
> Do you have a good or favorite proof or way to see it intuitively???
> I saw a cut and paste proof I liked in long ago in a book The Ascent
> of Man. But still I dont "Grok" (Heinlein) it.
>
> Maybe an algebraic proof would be better. Like...
> "Certaintly a+b=h (n=1) can't be true, so how about a^n + b^n = h^n.
> For n large, it certaintly cant be true.
> Now there is a right angle, which divides the plane into 4 equal
> regions...." I give up.
>
> And what about the generalization to non-right angles triangles?
> (Law of cosines I think)
> Have you "connected" that to anything?
>
> Got to go,
> Take Care,
> Dave