Sphere center from three tetrahedron face distances ($\mathbf{p}=r(|\mathbf{b}\times \mathbf{c}|\mathbf{a}+|\mathbf{c}\times \mathbf{a}|\mathbf{b}+|\mathbf{a}\times \mathbf{b}|\mathbf{c})/\tau$)

For a positively oriented basis $\mathbf{a},\mathbf{b},\mathbf{c}$, put $\tau =\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})>0$. The center of an interior sphere at distance $r$ from each of the three tetrahedron face planes through the origin has the displayed expression. The three inward unit vectors are the normalized cross products $\mathbf{b}\times \mathbf{c}$, $\mathbf{c}\times \mathbf{a}$, $\mathbf{a}\times \mathbf{b}$. Taking inner products with these vectors solves three independent signed-distance equations. Reversing some signs instead produces centers on exterior sides. Existence of a sphere contained in the full tetrahedron also requires clearance from the fourth face.