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Sep 28, 2005, 1:01:12 AM9/28/05

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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.

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.

It is the kind of thing that can be taught even to children. Yet it is

new.

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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Sep 28, 2005, 8:05:48 PM9/28/05

to

<charle...@hotmail.com> wrote in message

news:1127826603.6...@g49g2000cwa.googlegroups.com...

>

> 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.

Sep 29, 2005, 9:42:16 PM9/29/05

to

On Wed, 28 Sep 2005 05:01:12 GMT, charle...@hotmail.com wrote:

>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?

The perimeter of a square drawn on the side is four times the length

of that side?

You'd be better perhaps to consider still areas: Areas of *any*

similar figures drawn similarly on the sides of a right triangle bear

the same Pythagorean relationship as do the squares; the sum of the

two smaller add to the larger. Also, it is useful to know that in

similar figures linear measures are in the same ratio as the square

roots of the areas. That can be useful in so many ways.

So, you can draw similar triangles, or semi-circles, or whatever [even

very odd-looking, but exactly similar closed shapes] on the sides of a

right triangle, and the sum of the areas of the smaller will add to

the larger. Then there will also be a relationship between the

perimeters since the sides of the triangle are each the same factor of

the total perimeter of that similar figure. For example, the side of

the triangle is a diameter of a semi-circle drawn on that side, and is

so 2:(2+Pi) of that shape's perimeter, large or small.

I'd be careful of what I'd teach to children and what I'd leave for

later examination in a fuller context. Children have at first enough

difficulty with the normal Pythagorean theorm as it is usually taught.

Some discoveries are exciting to you because you already have the

necessary background to compare and contrast. To them, it could be

something the teacher wants, so they will do it, but it might have

little or no meaning to them out of a larger context.

Sep 29, 2005, 9:42:17 PM9/29/05

to

W H G wrote:

> Why teach it to children? New? maybe.

1. Why teach it to children? I said EVEN to children. It is not good to

underestimate children, and those who can understand Pythagoras can

understand this perimeter classification.

2. New? New to my understanding, and to those of my friends and

associates. However, such is the vastness of time and space that we

cannot know whether it was discovered before, unless a student of

Diophantus, or of Euclid's "Elements", or of Chinese or Indian

mathematical history finds the prior art.

Sep 30, 2005, 6:01:23 PM9/30/05

to

"Guess who" wrote:

>

> The perimeter of a square drawn on the side is four times the length

> of that side?

A square has four identical sides.

>

> You'd be better perhaps to consider still areas: Areas of *any*

> similar figures drawn similarly on the sides of a right triangle bear

> the same Pythagorean relationship as do the squares; the sum of the

> two smaller add to the larger. Also, it is useful to know that in

> similar figures linear measures are in the same ratio as the square

> roots of the areas. That can be useful in so many ways.

This is off topic, and adds nothing new.

I do not wish to guess who wrote this and more.

WHG, however, was completely in touch with what I am describing.

Perhaps I should give a few more extensions to the Pythagorean

Perimeter theorem?

First, the "SQUARE TRIANGLE" is 3 4 5. That is because a square on side

3 matches the perimeter of the triangle.

Secondly, the "TWO-TO-ONE" triangle is 5 12 13 because the perimeter of

a 10:5 rectangle matches the perimeter of the 5 12 13 triangle.

INVERSIONS.

I shall call these "inversions" because it is a term in the mathematics

of music. An octave UP is 2:1, whilst an octave DOWN is 1:2. That is

musical inversion - the ratio is stood on its head.

To invert 2:1, we use the ratio 1:2.

As before, the system is Diophantine, and uses no algebra because

Pythagoras lived about 1300 years before al Kwarismi, who introduced

algebra.

1. Take the ratio 1:2

2. Double it 2:2

3. Increment it 4:2 First side

4. Square it 16:4

5. Decrement it 12:4

6. Halve it 6:4 Second side

7. Increment it 10:4 Third side

8. Eliminate common denominator 4 3 5

So we have the 3 4 5 triangle reused as a 1:2 triangle (4 3 5).

The 1:2 rectangle on side 4 will have sides 2 and 4, giving a perimeter

12. That matches that of a square on side 3.

We can see that two ratios will map to each Pythagorean triangle - but

the ratios will not be the reciprocals of one another.

EXCHANGES

Here I propose to exchange the rectangles on the non-hypotenuse sides.

We have seen how the 3 4 5 triangle gives a 1:1 ratio whilst the 4 3 5

triangle gives a 1:2 ratio.

This theme - the exchanging of the rectangle from one side of the

right-angle to the other, and so halving or doubling the ratio -

repeats.

HYPOTENUSE

Here I propose that Pythagorean triangles may equally easily be

catalogued by matching them to Diophantine rectangles on the

hypotenuse.

Lengthy studies of these things would be boring. But I have to mention

them to show that I have studied them.

Charles Douglas Wehner

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