<s>[INST] What is the simplest proof of the pythagorean theorem you know? Maybe enough so to explain it to children. 
 [/INST] Consider the right figure and ignore the left one!

The area of the square of side $a+b$ is 
\begin{align*}
(a+b)(a+b) 
&= a^2 + ab + ab + b^2\\
&= a^2 + 2ab + b^2
\end{align*}
It is equal to the sum of 4 triangles and 1 square. This sum is
\begin{gather*}
4\times \frac{ab}{2} + c^2\\
2ab+ c^2
\end{gather*}
By equating both, we have
\begin{align*}
a^2 + 2ab + b^2
&= 2ab +c^2\\
a^2+b^2=c^2
\end{align*}
Now consider a single triangle so you have proven that $a^2+b^2=c^2$. Done!
</s>