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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