If you expand your final equation $$(2xy)^2 - (x^2 + y^2 - z^2)^2 = (2zx)^2 - (z^2 + x^2 - y^2)^2$$ you will get an equation which has equal terms in $x^4$, $y^4$, and $z^4$ on both sides. Removing these gets you:
$$2x^2y^2 + 2x^2z^2 + 2y^2z^2 = 2x^2y^2 + 2x^2z^2 + 2y^2z^2$$
which as you can see is always true.
The problem is that all your algebraic manipulations cannot add information, they can just rearrange what you already know. Look at your diagram. You have two sides of a triangle, and nothing else. There are an infinite number of triangles you can make that have two sides of given lengths. You can see this physically - get two sticks, attach their ends, and pivot. Every position gives you a new triangle, with a different third side length.