Center of Mass

Center of mass or barycenter (2 bodies):

$r_c$ is distance of center from $m_1$ along line between bodies

$r$ is distance between bodies

$m$ is masses

$$r_c=r\frac{m_2}{m_1+m_2}$$

Center of mass or barycenter (n>2 bodies):

(Sum of product of all masses and their associated positions)

$$\overrightarrow{r_c}=\frac{\Sigma \vec{p}m}{\Sigma m}$$

Mechanical Equilibrium

$\Sigma F$ Must equal zero to maintain linear state

$\Sigma \tau$ about any point must equal zero to maintain rotational state

If you put the point you are calculating at one of the forces, there's less algebra