<charleswehner@[EMAIL PROTECTED]
> wrote in message
news:1127826603.654689.318750@[EMAIL PROTECTED]
>
> It is not often that a new theorem of exquisite simplicity arrives -
> least of all in the context of Pythagoras.
>
> The world is vast, and time has aeons - so perhaps some other has made
> the same discovery.
>
> However, to me, and everyone I know this is a new discovery.
>
> It begins with the 3 4 5 triangle.
>
> You know - you create a square on side 3, add the square on side 4 and
> get the square on side 5. This is all to do with AREAS.
>
> However, what if you examine the PERIMETER of the square on side 3, and
> compare it with the perimeter of the triangle?
>
> What of the 5 12 13 triangle?
>
> Create a 2:1 rectangle on side 5. That is a 10:5 rectangle. How do the
> perimeters compare?
>
> What of the 7 24 25 triangle?
>
> Try a 3:1 rectangle - 21:7.
>
> And so on.
>
> These are the base triangles - the smallest.
>
The OP considers right triangles with the hypotenuse one more than
the longer side. Let a < b < c and a, b and c are a Pythagarian triple,
such that c = b +1 then
a^2 + b^2 = c^2 or a^2 + b^2 = (b+1)^2,
clear the parantheses and you can show
b = (a^2 - 1) / 2
If you choose an odd number for a then b will be an integer.
The perimeter is p = a+b+c = a + (a^2-1)/2 + (a^2-1)/2 + 1
so p = a^2 +a = a(a+1)
The rectangle with same perimeter is a by, call it, s
p = 2a + 2s so s = (p - 2a)/2 = ( a(a+1) -2a ) /2
and s = a(a-1)/2
Interesting, eh? The perimeter of the rt triangle is the same as
that of a rectangle with the shortest side (and another
integer side)! Indeed the side is determined by a.
> Of course, for the square you can scale up the 3 4 5 to 6 8 10, to 9 12
> 15 and onward.
>
> There are also 3:2 rectangles that convert to triangles. Indeed all
> whole-number ratios relate to base triangles.
>
Indeed there are. Consider the 8 15 17 triangle. Perimeter is 40,
same as a 8 by 12 rectangle with ratio of sides 3/2.
As above we can show that b = (a/2)^2 -1, a must be chosen
as an even such that b is odd or else the triangle is double one
of the triplets with c = b+1.
I'll let someone else do the algebra to derive the length of the
rectangle side.
> It is the kind of thing that can be taught even to children. Yet it is
> new.
>
Why teach it to children? New? maybe.
> Go to http://wehner.org/pythag
for the introduction, and follow the
> link, or directly to http://wehner.org/pythag/ratios.htm
for more
> examples of this perimeter-classification of right triangles.
>
> Charles Douglas Wehner
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