A graphical proof of the "Binomische Formeln"
A quick geometric proof of the three formulas $(a\pm b)^2=a^2 \pm 2ab + b^2$ and $(a+b)(a-b)=a^2+b^2$. These three can of course easily be proven by algebraic means, but it is nice to view them in a geometric context. As an aside on mathematical education, the three formulas are a set piece of high school education in Germany under the name "Binomische Formeln," but it appears that they do not have a similar status internationally.
The first two
The proof is simple, for the first "Binomische Formel" $(a+b)^2=a^2+2\,a\,b+b^2$ on the left hand side, we see directly that the square with sides of length $(a+b)$ is composed of a square with side $a$, one with side $b$ and two rectangles, each with one side $a$ and the other $b$ and adding together areas of the four rectangles gives us directly the result.
For the second one on the right, we have a square with side length $(a-b)$, the upper right square. To get from a square with sides $a$ to the one with sides $(a-b)$, we have to subtract the red rectangles, with sides $a$ and $b$, but then we have subtracted the dark blue square with sides $b$ in the lower left corner twice, and therefore we have to add it back.
Third square formula
The third Binomische Formel is $(a+b)(a-b)=a^2-b^2$ and the geometric proof is slightly less straight forward. We start with the rectangle $II$, with the sides $(a+b)$ and $(a-b)$ and try to construct the square with sides $a$, the black outline. For that we note first, that the rectangle $II$ decomposes into two parts, one the rectangle with sides $a$ and $(a-b)$ and second the rectangle $I$ at the bottom with sides $b$ and $(a-b)$. Then we move $I$ over to the left side and the missing piece is the square with side length $b$, shown in red.
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