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