Coordinates from Triangle Constraints

We need to solve the equations $|\mathbf{a}\times\mathbf{d}| \; = \; 4 \hspace{1cm}|\mathbf{b}\times\mathbf{d}| \; = \; 6 \hspace{1cm} |\mathbf{c}\times\mathbf{d}| \; =\; 8$ where $\mathbf{a} = \overrightarrow{OA}$ , $\mathbf{b} = \overrightarrow{OB}$ , $\mathbf{c} = \overrightarrow{OC}$ , $\mathbf{d} = \overrightarrow{OD}$ . If we suppose that $D$ has coordinates $(x,y,z)$ , we are led to the equations $\begin{aligned} 2x^2 + 2y^2 + 2z^2 - 2xy - 2xz - 2yz & = \; 16 \\ 5x^2 + 2y^2 + 5z^2 + 4xy + 2xz - 4yz & = \; 36 \\ x^2 + 5y^2 + 4z^2 - 4xz & = \; 64 \end{aligned}$ Solving these equations numerically, there are four real solutions, $(-0.876669, -3.57361, -0.629862) \hspace{1cm} (0.876669, 3.57361, 0.629862) \hspace{1cm} (0.0509774, 2.077, 3.28242) \hspace{1cm} (-0.0509774, -2.077, -3.28242)$ and only the second of these satisfies the requirement that $x,y,z > 0.1$ . Thus the answer is $\lfloor 1000(0.876669 + 3.57361 + 0.629862)\rfloor \; = \; \boxed{5080}$