## Most recent change of PythagorasTheorem

Edit made on November 12, 2009 by DerekCouzens at 08:43:55

Deleted text in red / Inserted text in green

WW
Pythagoras' Theorem states that in a right angled triangle, the sum of the squares of the lengths
of the two shorter sides is equal to the square of the length of the longest side (the hypotenuse).

[[[> IMG:pythagoras.png ]]]

There are literally hundreds of proofs of this theorem, including one found/created by James A. Garfield,
who later became US president.

Albert Einstein also discovered a proof which is demonstrated at: http://demonstrations.wolfram.com/EinsteinsMostExcellentProof/

What is less commonly known is that this is an "if and only if."

Consider a triangle *T* with sides a, b and c, with c the longest. Stating both parts:

* If *T* is a right angled triangle, then EQN:a^2+b^2=c^2,
* If EQN:a^2+b^2=c^2, then *T* is a right angled triangle.

The fact that a 3:4:5 triangle has a right angle was certainly known to the ancient Egyptians, and was
used by their builders.

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!! WARNING: Incomplete advanced material follows ...
Here is a proof that the Greeks would never have accepted.

Consider a right-angled triangle. Draw the triangle on the complex plane with the hypotenuse running from
the origin into the first quadrant, and the right angle on the X-axis. The vertices of the triangle are
now at 0, /a+0i/, and /a+bi./ Using Euler we can write EQN:a+bi=ce^{i\theta}.

Take the complex conjugate, and multiply. That gives us
| EQN:(a+bi)(a-bi)=(ce^{i\theta})(ce^{-i\theta}) |
which simplifies to
| EQN:a^2+b^2=c^2 |
and we're done for the first direction.

Most of this is reversible, so there's very little to check for the other direction.
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