Pythagorean Proof Based on the Principles of Scaling

Date: 04/04/2002 at 00:39:38
From: Charlene
Subject: Innovative ideas for Pythagorean Theorem

Hi Dr. Math,

I'm a ninth-grader looking for help for a science fair project. I've 
decided to do a project with some connections to the Pythagorean 
theorem, but the project requires innovative ideas. The following are 
my current ideas for the project: 

a) Find several proofs of the theorem.
b) Develop methods for finding pythagorean triples. (I already know 
   of four, two of which I discovered myself)
c) Investigate Fermat's Last Theorem. (It does have some connections 
   with the Pythagorean Theorem, doesn't it?)

Please, could you give me more ideas?

Thank you very much for your help.

Date: 04/04/2002 at 00:55:31
From: Doctor Mitteldorf
Subject: Re: Innovative ideas for Pythagorean Theorem

I don't think you need more ideas, Charlene. You already have a bunch 
of good ones. You can start reading about the theorem in the Dr. Math 
FAQ:

   Pythagorean Theorem
   http://mathforum.org/dr.math/faq/faq.pythagorean.html   

Here are a few sites on the Web that prove the Pythagorean Theorem, 
many with pictures, some with more detail than others. Most involve 
areas in some form. Among these you should probably find at least one 
that is explained for you.

Euclid's proof (with notes on two alternative proofs) - David Joyce
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html   

An Interactive Proof of Pythagoras' theorem (Euclid's proof in Java)
- Jim Morey, UBC Mathematics Department
http://sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html   

A proof of the Pythagorean Theorem by Liu Hui (third century AD)
- D. Wagner, University of Copenhagen
http://www.staff.hum.ku.dk/dbwagner/Pythagoras/Pythagoras.html   

Pythagorean Theorem (Java applets) - Manipula Math
http://www.ies.co.jp/math/java/geo/pythagoras.html   

38 proofs and some more references - Cut-the-Knot, Bogomolny
http://www.cut-the-knot.org/pythagoras/index.html   

The Pythagorean Theorem - Jim Loy  
http://www.jimloy.com/geometry/pythag.htm   

There are also several explanations in the Dr. Math archives, 
including these; the last proof Dr. Rob gives fits your description 
well, as does Dr. Mitteldorf's, which includes both picture and 
explanation:

   Proving the Pythagorean Theorem: A Traditional and a Modern Approach
   http://mathforum.org/library/drmath/view/54986.html

   The Pythagorean Theorem: A Modern Proof
   http://mathforum.org/library/drmath/view/54711.html   

And here's one that was new to me a few weeks ago. I like this proof 
because it seems to give you "something for nothing."  There isn't any 
calculation in it, and it's based completely on the principles of 
scaling.

Draw your right triangle with the hypotenuse as base. Let's call it 
ABC, with AB being the base and C being the point on top. Draw the 
altitude from C, making a right angle with the base.

  

The altitude divides the triangle into two parts, so the areas of 
those two parts must add up to the total area. Now, it turns out that 
all three triangles - the big one and the two parts - are similar.  
"Similar" means they are the same shape, but different sizes.  If all 
the angles match, then the triangles are similar. Can you prove that 
the angles match up in these three triangles?

For the next step in the proof, you need a "scaling" principle. The
principle says that if you take any shape and "scale it up" so that 
every line in it is f times bigger, then the area will be larger by 
f^2.

To state the scaling principle in another way: the area of a figure is
proportional to the square of any of its linear dimensions. So if any 
line in the figure has a length = d, then the area will be r*d^2, and 
the r will stay the same if you change d, so long as you change the 
whole figure proportionately.

Now, the hypotenuse of the big triangle is c, and the hypotenuses of 
the parts are a and b. So the areas are rc^2 for the whole triangle, 
and ra^2 and rb^2 for the two parts. The parts add up to the whole, 
so ra^2+rb^2=rc^2. Then you can just divide both sides of the equation 
by r.

Or, using the first statement of the scaling principle, if we take the 
big triangle as standard, with f = 1, then the "f" for one triangle is 
a/c and for the other it is b/c. The first scaling principle then 
tells us that (a/c)^2 + (b/c)^2 = 1.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/