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I have been looking for proofs for the pythagorean theorem that don't use area calculation but calculus, complex numbers or any other interesting ways to proof it.

I would love to see any interesting proof, Shay

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    I can not be anymore specific than that. Any axiom, b$u$t those that define distance as I said, is ok.2011-07-22

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Explanation in terms of linear algebra. From this blog post by Terence Tao

The statement $a^{2}+b^{2}=c^{2}$ is equivalent to the assertion that the matrices $% \begin{pmatrix} a & b \\ -b & a% \end{pmatrix}% $ and $% \begin{pmatrix} c & 0 \\ 0 & c% \end{pmatrix}% $ have the same determinant. But it is easy to see geometrically that the linear transformations associated to these matrices differ by a rotation, and the claim follows.

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    @shayfalador: You followed up this subject!2012-03-09
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There is a proof using Similar Triangles:

enter image description here

We get from $\triangle CDA$ and $\triangle ABC$ that $\displaystyle \frac{CD}{AC} = \frac{CA}{BC}$ i.e. $\displaystyle \frac{\alpha}{a} = \frac{a}{c}$ i.e. $ \alpha c = a^2$

Similarly $\displaystyle \beta c = b^2$.

Adding gives the result.

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    By the way: here's my preferred way of presenting this proof. [Look at this picture:](http://i.stack.imgur.com/crYRD.png) the red, green and blue figures are similar "houses". Since the red and blue roofs give the green roof, the red square and the blue square must add up to the green one. QED --- [Christian Blatter](http://math.stackexchange.com/users/1303/christian-blatter) displayed a similar picture with the heading: "Ich bin auch ein Beweis!" ("I'm a proof, too" in German) at his fare-well lecture. [There is a joke in that title that is hard to explain to people not living in Zurich]2011-07-22
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Isn't the answer is just what's in here: Proof using differentials ?

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    @ItsNotObvious but it is a fraction...2016-06-28
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How about the simplest ever (I got it from a book): Imagine a right prism with the base the triangle in question A(right angle), B , C (counter clock wise). The height of the prism is arbitrary. Now fill the prism with a gas at a given pressure. On the faces, in their middle the pressure forces act (Surface * pressure, perpendicular on the face). Equate the momenta around the corner B. They should cancel (the prism does not rotate by itself). You get the Pitagora's theorem in a blink of an eye. Cheers!! PS: I wanted to upload the figure, but I am not allowed until I get more points!

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    If you put the figure somewhere I c$a$n see it, I will upload it for you.2012-06-05
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http://www.cut-the-knot.org/pythagoras/CalculusProofCorrectedVersion.shtml

This proof also appears in The American Mathematical Monthly, April 2011 issue.

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An elegant proof. Consider the vector $\mathbf{c}$ in the complex plane: $\mathbf{c}= a + \mathbf{i}b$, where $a$ and $b$ are scalar values, and $a,b > 0$. Now consider the reflection of $\mathbf{c}$ in the line at an angle of 45 degrees (pi/4 radians) passing through the origin, call that $\mathbf{d}$. $\mathbf{d} = b + \mathbf{i}a$, and $\mathbf{d}$ clearly has the same length as $\mathbf{c}$.

Now take the vector product of c and d. That has the same length as the square of the length of c, and in terms of polar coordinates, it is at an angle of pi radians (by virtue of the multiplicative properties of complex numbers expressed as polar coordinates).

so $\mathbf{c\cdot d} = i(c^2) = (a+ib)\cdot (b+ia) = ab-ba+i(a^2 + b^2) = i(a^2 + b^2)$.

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