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).

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 $a^2+b^2=c^2,$
If $a^2+b^2=c^2,$ 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.

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's polar representation of a complex number we can write $a+bi=ce^{i\theta}.$

Take the complex conjugate, and multiply. That gives us

$(a+bi)(a-bi)=(ce^{i\theta})(ce^{-i\theta})$

which simplifies (using the difference of two squares) to

$a^2+b^2=c^2$

and we're done for the first direction.

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

See also:

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

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

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Pythagoras Theorem

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WARNING: Incomplete advanced material follows ...

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