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### 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/
```
Associated Topics:
College Coordinate Plane Geometry

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