Section Formula

Point dividing PQ internally in ratio m:n. For external division, replace n with −n. Midpoint is the special case m = n = 1.

Let $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$. Let $R$ divide $PQ$ internally in ratio $m:n$, so $PR:RQ = m:n$.

x-coordinate: Project onto the x-axis. Lengths $PR$ and $RQ$ project to $x_R-x_1$ and $x_2-x_R$. Since $PR:RQ = m:n$:

$$\frac{x_R-x_1}{x_2-x_R} = \frac{m}{n} \implies n(x_R-x_1) = m(x_2-x_R)$$ $$x_R(m+n) = mx_2+nx_1 \implies x_R = \frac{mx_2+nx_1}{m+n}$$

Similarly for $y$ and $z$.

$$R = \left(\frac{mx_2+nx_1}{m+n},\;\frac{my_2+ny_1}{m+n},\;\frac{mz_2+nz_1}{m+n}\right)$$

External division (ratio $m:n$ externally): Replace $n$ with $-n$:

$$R = \left(\frac{mx_2-nx_1}{m-n},\;\frac{my_2-ny_1}{m-n},\;\frac{mz_2-nz_1}{m-n}\right)$$

Midpoint ($m = n = 1$):

$$M = \left(\frac{x_1+x_2}{2},\;\frac{y_1+y_2}{2},\;\frac{z_1+z_2}{2}\right)$$

Centroid of triangle with vertices $A$, $B$, $C$:

$$G = \left(\frac{x_1+x_2+x_3}{3},\;\frac{y_1+y_2+y_3}{3},\;\frac{z_1+z_2+z_3}{3}\right)$$