Just let it flow

Find the flux of the vector field $\vec{F}(\vec{r})=\frac{1}{\pi}\frac{\vec{r}}{||\vec{r}||^3}$ through the triangular surface $S$ with its vertices at $(1,\phi,0),(0,1,\phi)$ and $(\phi,0,1)$ , where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio.