If you look at the bottom of the page

https://www.cut-the-knot.org/pythagoras/euclid.shtml

you'll see that Scott Brodie had exactly same diagram as yours but short of the semicircle. So it appears that your geometry is a particular case of Euclid VI.31.

However, your reasoning that explicitly starts with the semicircle and a tangent, appears to me sufficiently independent to warrant its inclusion as a separate proof. I'll do that a short time later.

So I do thank you for sending this to me.

All the best,
Alexander Bogomolny