Euler line

We wish to show that for any triangle, $HG=2GO$. To do this, notice that we can interpolate between the altitudes and perpendicular bisectors to get the median. Let $l$ be the line parallel to $h_A\text{and}{\perp }_{BC}$ that passes through $G$. Then notice that this is $\frac{2}{3}$ of the way from $h_A$ to ${\perp }_{BC}$. Therefore, $G$ is $\frac{2}{3}$ of the way from $H$ to $O$, completing our proof.