Well done to Nayanika from The Tiffin Girls' School, Yihuan form Pate's Grammar School, John from Calthorpe Park School in the UK and Andrew from Island School, who all sent in correct proofs.

This is John's work:

The four triangles are similar by AA similarity (they all share two angles - and therefore all 3 angles). As triangle 4 is similar to triangle 1 its corresponding parts are in equal ratios.

$\therefore a:ac$ is the same ratio as $c:a^2+b^2$

$\therefore 1:c$ $=$ $c:a^2+b^2$

$\therefore 1:c$ $=$ $1:\dfrac{a^2+b^2}{c}$

$\therefore c = \dfrac{a^2+b^2}{c}$

Which is rearranged to $a^2+b^2=c^2$

Alternatively we can think of triangle 4 as triangle 1 enlarged by a scale factor of $c$.

Therefore $a^2+b^2 = c\times c$

So $a^2+b^2=c^2$