biggest square inside a right angled triangle

maybe an interesting application of similar triangle techniques
but some hefty algebraic manipulation is needed to establish a general result

students could possibly be presented with the derivation of a general result and be asked to sort out what is happening at each stage

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