What is the simplest proof of the pythagorean theorem you know? Maybe enough so to explain it to children. 

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!
