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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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See also:

* http://mathworld.wolfram.com/PythagoreanTheorem.html

** Note that this site only deals with one direction of the theorem.