UPDATE Making the Equilateral Triangle on the Parabola y=x^2 a 3, 4, 5 Triangle

I failed. It's almost as if Dr Munro had predicted it as I (thankfully) found this solution:
https://www.maths.ox.ac.uk/system/files/attachments/Interview_Question_2019_JM.pdf

Key mistakes I was making:

1. As it had to be similar to a 3, 4, 5 triangle I was equating equations to a certain 3λ, 4λ and 5λ. Introducing another variable into the mix was not brilliant so the idea of representing each side as a proportion of the other was a lot cleverer.
2. I somehow managed to forget that two perpendicular lines have gradients that multiply to make -1. That would've been a good piece of information to know (!)

Anyway, I have a series of questions from IWantToStudyEngineering that I did on Sunday that shall be written up (and possibly typed up) soon. For now I will be reconsidering my application to Oxford because that wasn't even as hard as I made it out to be.