I think if you’re not too picky about what counts as showing something, then there’s very little that can’t be given some kind of an explanation. I think some volume formulae might be the only things

Depends on how picky you want to be with what counts as showing something is true. You can pretty much always go deeper on the why, in which case it might never be the case (e.g measure theory before area/volume, peano axioms before adding etc)

And overall difficulty is of course definitely above what would ever be expected at GCSE

I think calling it 'just-about GCSE level' was a bit too much of a stretch! Thought was that something like 'prove that the square of an odd number is one more than a multiple of 4' doesn't require any mod arithmetic heavy-lifting.

The most surprising thing doing this was what the product of the gradients of the tangents is in terms of a, b and c

It's from the textbook for Edexcel's new L2 Extended Maths Certificate

Thanks

I’d be curious about reading this. Do you have a reference? Thanks

Very nice - love a proof by folding

And an animation of it in geogebra, because why not