I remember a lecturer of mine having to state and prove a theorem due to Heawood[1]. I think he did it first because the statement was quite unpleasant and not easy to memorise. I found the point of proving things from memory is that to memorise a proof, one must distill it to key ideas with obvious steps in between. This was particularly useful if you might need to reproduce a proof in an exam.

[1] https://en.m.wikipedia.org/wiki/Heawood_number