My conclusion was actually that the points M and N must be the same point, which then implies that they also coincide with R.

The drawing can be "faked" by sketching the diameters to be a bit off-center, which results in M,N and R not being exactly the same point.

No ?

Exactly.

I don't see the point of this. It's wrong, if you spend a moment's thought then it's obvious why it's wrong, and it's about as enlightening or entertaining as the usual proof that 0=1.

Can anyone explain what they like about this?