Given the coordinates of $3$ vertices forming a triangular face of an irregular octahedron, and the lengths of the $9$ edges which meet at the $3$ unknown vertices, I wish to know the coordinates of these unknown vertices. I am able to find iterative solutions but I feel that this is a determinate structure and therefore a closed solution should be possible for this problem. This solution would be useful in the analysis of structures and measurement networks involving octahedra and tessellations of octahedra and tetrahedra (octet truss).
Does anyone have a solution? My preferred language for this would Matlab.