William_Thompson

2005-Dec-25, 10:34 PM

When I was a kid, one of my most favorite things to do was to solve mysteries. If there was a magician on television, I got more enjoyment figuring out how the trick was done than being amazed by the trick itself.

There was elation in figuring something out and I always wanted to share it with people.

When I learned of the Pythagorean Theorem I was taught that it was true simply because regardless of the dimension of the right angled triangle, the hypotenuse squared always equaled the sum of the other two sides squared. But when asked why, the teacher simply said that it just happens to always work out that way and that there wasn’t really any proof written.

http://www.completetranslations.com/christmas_math/one.JPG

Well, I took it upon myself to find one. And I did. And as a gift I am going to post it here now.

There are two interesting observations about a right angle triangle. obviously, you can make a rectangle with two identical right triangles put together at their hypotenuse and the area of this rectangle would be a∙b.

http://www.completetranslations.com/christmas_math/two.JPG

What is less obvious is that if you take the two identical rectangles from the first observation and make a square by using the hypotenuses as the parameter and the area of this square would be c². This square form the second observation would have a square inside of it whose side would measure b-a and so its area is (b-a)².

http://www.completetranslations.com/christmas_math/three.JPG

http://www.completetranslations.com/christmas_math/four.JPG

No matter what proportion of right angled triangle you use, the result is always that you can form a perfect square with four identical triangles using their hypotenuse and you will always have a little empty square in the middle. I find it interesting conceptualizing this as you go through the possibilities, the little empty square in the middle spins and collapses like the iris on a camera.

Regardless of the size of the empty square formed in the middile of the square whose sides are of lenght "c". and regardless of the proportions of the 4 identical triangles, the length of each side of the inner square will always be b-a.

I call the area of the whole square, "Area C" where each side of the square is made of the hypotenuse of 4 identical triangles.

And the square area of the empty space inside this square I will call "Area B".

And the area formed by the two rectangles to be called "Area A".

The proof is simple algebra and substitution using the equations determined:

http://www.completetranslations.com/christmas_math/five.JPG

There was elation in figuring something out and I always wanted to share it with people.

When I learned of the Pythagorean Theorem I was taught that it was true simply because regardless of the dimension of the right angled triangle, the hypotenuse squared always equaled the sum of the other two sides squared. But when asked why, the teacher simply said that it just happens to always work out that way and that there wasn’t really any proof written.

http://www.completetranslations.com/christmas_math/one.JPG

Well, I took it upon myself to find one. And I did. And as a gift I am going to post it here now.

There are two interesting observations about a right angle triangle. obviously, you can make a rectangle with two identical right triangles put together at their hypotenuse and the area of this rectangle would be a∙b.

http://www.completetranslations.com/christmas_math/two.JPG

What is less obvious is that if you take the two identical rectangles from the first observation and make a square by using the hypotenuses as the parameter and the area of this square would be c². This square form the second observation would have a square inside of it whose side would measure b-a and so its area is (b-a)².

http://www.completetranslations.com/christmas_math/three.JPG

http://www.completetranslations.com/christmas_math/four.JPG

No matter what proportion of right angled triangle you use, the result is always that you can form a perfect square with four identical triangles using their hypotenuse and you will always have a little empty square in the middle. I find it interesting conceptualizing this as you go through the possibilities, the little empty square in the middle spins and collapses like the iris on a camera.

Regardless of the size of the empty square formed in the middile of the square whose sides are of lenght "c". and regardless of the proportions of the 4 identical triangles, the length of each side of the inner square will always be b-a.

I call the area of the whole square, "Area C" where each side of the square is made of the hypotenuse of 4 identical triangles.

And the square area of the empty space inside this square I will call "Area B".

And the area formed by the two rectangles to be called "Area A".

The proof is simple algebra and substitution using the equations determined:

http://www.completetranslations.com/christmas_math/five.JPG