Coordinate/Analytic Geometry and the Pythagorean Theorem

Date: 6/11/96 at 9:36:49
From: Anonymous
Subject: Pythagoras' theorem

I am a mechanical engineer from Sweden. I have a simple question on 
coordinate geometry (or analytic geometry).

How can you prove Pythagoras' theorem with pure coordinate methods?
Usually the concept "distance" is introduced in coordinate geometry by
using Pythagoras' theorem, but I want to prove it with the methods of
coordinate geometry.

Thank you
John Engel

Date: 9/5/96 at 12:19:20
From: Doctor Ceeks
Subject: Re: Pythagoras' theorem

Hi,

The Pythagorean theorem was most likely originally intended as a 
statement about the areas of squares erected on the sides of a right 
triangle.

If you view the theorem in this manner, then to prove it using 
coordinate geometry you would have to define notions of area and also 
of angle, so that you can speak of "right" triangles.  Doing so would 
probably amount to a translation of the existing proofs.

The notion of distance is tricky, and you're right that Euclidean 
distance is very often built into the definition of coordinate 
geometry.

However, it is well known now that Euclidean distance isn't the only 
meaningful distance to define on the coordinate plane (= pairs of 
reals (x,y)).

So the answer to your question depends on what you mean by "pure 
coordinate methods".

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/