Sylvester's triangle problem

Theorem

Consider three pairwise distinct vectors of equal length ${\displaystyle \overrightarrow{u}}$, ${\displaystyle \overrightarrow{v}}$ and ${\displaystyle \overrightarrow{w}}$ each of them acting on the same point ${\displaystyle O}$ thus creating the points ${\displaystyle A}$, ${\displaystyle B}$ and ${\displaystyle C}$. Those points form the triangle ${\displaystyle \mathrm{△}ABC}$ with ${\displaystyle O}$ as the center of its circumcircle. Now let ${\displaystyle H}$ denote the orthocenter of the triangle, then connection vector ${\displaystyle \overrightarrow{OH}}$ is equal to the sum of the three vectors:

${\displaystyle \overrightarrow{OH}=\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}}$

Furthermore, since the points ${\displaystyle O}$ and ${\displaystyle H}$ are located on the Euler line together with the centroid ${\displaystyle S}$ the following equation holds:

${\displaystyle \overrightarrow{OH}=3\cdot \overrightarrow{OS}}$