Past exam of the mathematics course of the University of Cambridge / 2002 / ia / Paper 3 / 8B (Algebra and Geometry)

The three known vertices have pairwise squared distances $8$, so the regular tetrahedron must have every squared edge length equal to $8$. Comparing the squared distance from $d$ to $b$ with that from $d$ to $a$ givesEquality forces $x=-1$. Similarly,so $y=-1$. The resulting point has squared distance $8$ from each of the other three vertices. Therefore

Take the half-turn about the $z$-axis, represented by the rotation matrixIt is an orthogonal matrix with determinant $1$, and it fixes the $z$-axis while reversing both perpendicular coordinates. Direct substitution gives $Ma=b$, $Mb=a$, $Mc=d$ and $Md=c$. A linear map that permutes the four vertices also preserves their convex hull, which is the solid tetrahedron. Hence this rotation matrix leaves $T$ invariant. Its axis joins the midpoints of the two opposite edges $ab$ and $cd$.