Incenter of a tetrahedron ($I=(S_{A}A+S_{B}B+S_{C}C+S_{D}D)/(S_A+S_B+S_C+S_D)$)

The center of the sphere tangent to all four faces is the positive face-area-weighted combination of the opposite vertices. If $V$ is the volume and $S$ the total face area, its distance from each face is $3V/S$. Indeed its weight for vertex $A$ is $S_A/S$ and the corresponding altitude is $3V/S_A$, so their product is the common distance. Positive weights place it inside the tetrahedron.