Right, so this was something I stumbled upon whilst looking at forms of the golden ratio (if you replace x with one in the diagram above then you get phi) and then I wondered about what if you messed with the value of x, did a massive amount of trial and error (lost in a notebook somewhere) then came up with the equation in the right, which worked exactly for every value I tested (about up to 100)
Now I have found stack exchange, in wondering if there is a way to prove this, I plugged in some values into wolfram alpha and geogebra, which seems to show it works, but I want a proof, not just trial and error answers.