Trying to find Sylvester's triangle problem proof

[MathNugget]

If ABC is a triangle, O is the intersection of the perpendicular bisectors, and H is the orthocenter (intersection of heights / altitudes) not so sure these are the proper words. (I attached a pdf with what I could find so far. it's part of a bigger exercise, but I am not following up the part where it proves what I want).

This is what I'd like to see how it's proven: $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{OH}$ . Proving that $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{ON} \Longrightarrow N = H$ doesn't really do it.

I can see that, alternatively, it's enough to prove $2 \overrightarrow{OU} = \overrightarrow{AH}$, where U is the middle of BC.

I found this result is Sylvester 's triangle problem. Still cannot fully understand if the proof is complete, or if maybe there's a way to prove those things without starting with the conclusion.