For reference: If ABCD and EFGH are squares, HI = 2 and GN=$\sqrt5$ , calculate EH.

[![enter image description here](https://i.stack.imgur.com/Ct0ZL.jpg)](https://i.stack.imgur.com/Ct0ZL.jpg)

My progress:

If point I were given as the midpoint of $AD$ the exercise would be quite easy $AE=4=AI, EH = \sqrt{(4^2+2^2)} = \sqrt20 = 2\sqrt5=GH$  

so I think the idea is to demonstrate that it is the midpoint. I drew some auxiliary lines...

[![enter image description here](https://i.stack.imgur.com/cUKGo.jpg)](https://i.stack.imgur.com/cUKGo.jpg)

[![enter image description here](https://i.stack.imgur.com/gTUYv.jpg)](https://i.stack.imgur.com/gTUYv.jpg)

[![enter image description here](https://i.stack.imgur.com/mwBow.jpg)](https://i.stack.imgur.com/mwBow.jpg)