Hard basic triangle problem

[pedrodu36]

Hi everyone, hope you're having a fantastic day.

I got stuck on a problem while playing some polybridge (picture related). The restrictions depicted in the drawing are correct (e.g. D is on a straight line between A and C).

<BAC=90° ; AB=3 ; BE=2 ; DE=2 ; CD=4 ; CE=4 ; AD=???

Have a crack at it, I couldn't solve it.

[pedrodu36]

Here is some of the progress I have made. It seems to me that it gets too complexe and there must be an easier way.

To keep the drawing clean, I have numbered each subtriangle and classified all the angles accordingly using the greek letter equivalent of the point (e.g. epsilon for E).

Worth noting that the subtriangle III is solved, so I thought about starting from there.

[Guest]

AD =[tex]\sqrt{3}[/tex] ??