Orthocentre Geometry Proof

Question

I need help with the following geometry problem:

Let $O$ be the centre of circumscribed circle of $\Delta ABC$ and $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be the vectors pointing from $O$ to their vertices.

Let $M$ be the endpoint of $\mathbf{a}+\mathbf{b}+\mathbf{c}$ measured from $O$.

Prove that $M$ is the orthocentre of $\Delta ABC$.

Answer 1

Note that $O'$ is orthocenter iff $$ (O'-c )\cdot (a-b)=0 \ \ast$$ for $c$ and for $a,\ b$ corresponded equation wrt $\ast$ hold

Note that if $H=a+b+c$, then $$(H-c) \cdot (a-b)=|a|^2-|b|^2=0$$ That is $H=O'$