Tetrahedral Law of Cosines Proof Given a tetrahedral $\rho$ with faces $\Xi, \Pi, \Gamma, \Delta$ with areas $\xi , \pi, \gamma , \delta$, respectively, assign a normal vector to each face such that $\mid \mid \hat{\xi} \mid \mid = \xi, \mid \mid \hat{\pi} \mid \mid = \pi, \mid \mid \hat{\gamma} \mid \mid = \gamma, \mid \mid \hat{\delta} \mid \mid = \delta$
Prove that the sum of the vectors is the zero using vector cross product.      
 A: I happened to draw some TikZ illustration for the some computational geometry slides, which may be handy in explaining. Just to continue Blue's argument in the comments.

Say $\newcommand{\b}{\boldsymbol}\b{a} = \vec{V_1 V_2}$, $\b{b} = \vec{V_1 V_4}$, $\b{c} = \vec{V_1 V_3}$. Then
$$
\b{a}\times \b{b}, \; \b{b}\times \b{c}, \; \b{c}\times\b{a}\tag{1}
$$
will produce the vectors normal to the triangular faces $F_3$, $F_2$, $F_4$ opposite to the vertex $V_3$, $V_2$, $V_4$ respectively, with length $2|F_3|$, $2|F_2|$, and $2|F_4|$ (absolute value just denotes the area). Also notice that all of them point outward with respect to this tetrahedron.
For the last vector normal to the face $F_1$ opposite to $V_1$, notice that 
$$
\vec{V_3 V_4} = \vec{V_1 V_4} - \vec{V_1 V_3} = \b{b}-\b{c},
\\
\vec{V_4 V_2} = \vec{V_1 V_2} - \vec{V_1 V_4} = \b{a}-\b{b}.
$$
Then the last vector $\b{n}_{F_1}$ is produced by 
$$
(\b{b}-\b{c}) \times (\b{a}-\b{b}).\tag{2}
$$
Adding everything factored by $1/2$ in (1) and (2), using the anti-commutativity of the cross product, see what happens. 

Another nice proof I like is to use the Barycentric coordinate function, the vector normal to a triangular face opposite to the vertex $V$, having the scale of the area of that face, happens to be a constant about the volume times the gradient of barycentric coordinate $\lambda_V$ of this very vertex $V$. $\lambda_V$ is a linear function valued $1$ at $V$, and linearly decreases to $0$ to other three vertices. The sum of the normal being zero is just the the gradient of the sum of all four barycentric coordinates being zero (barycentric coordinates sum to $1$). 
