anicca

Your labelling is a bit confusing, especially as D can't be both the point furthest from B and the common corner. If you replace that D with F and call the first square ABCD then it makes a bit more sense, and sketches make it look plausible. I've got quite a bit of work to do before a meeting this afternoon, but I'll have a go at proving it afterwards. It does look like the sort of thing Euclid would have done.