Pythagorean Theorem - Art Of Mathematics

Proof without words: The Pythagorean Theorem

Explanation

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.

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