Ok, I read through the proof, and I think I understood it. Thanks for posting this!
Until the authors' work is submitted to a journal and reviewed, it's hard to say everything claimed here is definitely correct & new.
Update: Nice video on the proof: https://www.youtube.com/watch?v=nQD6lDwFmCc I like what they did :) Seems legit to me.
Btw, law of sines can be proven independently of pythagoras. So using that as a step is ok. https://en.wikipedia.org/wiki/Law_of_sines