Sylvester's triangle problem

Consider three pairwise distinct vectors of equal length ${\displaystyle {\vec {u}}}$, ${\displaystyle {\vec {v}}}$ and ${\displaystyle {\vec {w}}}$ 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 ${\displaystyle \triangle ABC}$ 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:^[1][2]

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

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:^[3]

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