A New Proof of Pythagorean theorem?

This is really just an overly complex use of similar triangles to show$$m/a=h/b,\,n/b=h/a\implies c=m+n=(a^2+b^2)/c,$$where in the last $=$ I, like you, have used $ab=ch$. As with your other questions I've answered about Pythagorean proofs, the strategy works but is intimately related, once extraneous Cartesian coordinates have been stripped away, to old approaches. In particular, computing $x,\,y$ is equivalent to computing $m,\,n$, so there's no need to worry about how axes are oriented.