Solution
= Solution
Dotting $c=\sum_j\gamma_ja_j$ with $e_i$ gives $e_i\cdot c=\gamma_i\lambda_i$, proving $\gamma_i=(e_i\cdot c)/\lambda_i$. Also
$$
a_1\cdot(a_2\times a_3)=\frac{\lambda_1\lambda_2\lambda_3}{\Delta},\qquad
\gamma_1\gamma_2\gamma_3=-\frac{(c\cdot v+d)^3}{\lambda_1\lambda_2\lambda_3}.
$$
Combining these with $V=|\Delta|/6$ gives the displayed face formula, with the orientation chosen so its signed right-hand side is positive.