Are there any calculus/complex numbers/etc proofs of the pythagorean theorem?

Question

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

Answer 1

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.

Answer 2

There is a proof using Similar Triangles:

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.

Answer 3

Isn't the answer is just what's in here: Proof using differentials ?

Answer 4

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!