a good, if demanding, application of similar triangles
what are the steps?
unsurprisingly, I subsequently found that this has been explored by (amongst others) Marion Walter in FLM 21 in 2001 and MT 53 in 1970
her general results agree with those above (it looks like there is a small numerical error on p.29 of her article)
she points out that 1/y^2 = 1/x^2 + 1/c^2 where 'c' is the hypotenuse