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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:PythagorasTheorem.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. ---- !! 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 | EQN:(a+bi)(a-bi)=(ce^{i\theta})(ce^{-i\theta}) | which simplifies (using the difference of two squares) 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. ---- See also: * http://mathworld.wolfram.com/PythagoreanTheorem.html ** Note that this site only deals with one direction of the theorem.