Proof without words: The Pythagorean Theorem


We take a right triangle with sides \(a\), \(b\) and \(c\)
Then we duplicate the yellow triangle to get 8 similar triangles
Now we assemble the triangles to get the sides of a square like it is shown in the next figure

Next, let’s try to fill the white space until we get a filled square. \[\] We start by adding 2 blue squares of the same side within the previous figure. We get
Then, we add 2 green squares of side b
Finally, we fill the rest by adding the two last squares of the same side coloured in red
We can divide the area of the biggest square into these parts to get our final result.

\[\huge \left(2c\right)^{2}=2c^{2}+2a^{2}+2b^{2}\Rightarrow c^{2}=a^{2}+b^{2}\]