  We derive several results in classical Euclidean elementary geometry using
the steering ellipsoid formalism from quantum mechanics. This gives a
physically motivated derivation of very non-trivial geometric results, some of
which are entirely new. We consider a sphere of radius $r$ contained inside
another sphere of radius $R$, with the sphere centres separated by distance
$d$. When does there exist a nested tetrahedron circumscribed about the smaller
sphere and inscribed in the larger? We derive the Grace-Danielsson inequality
$d^2 \leq (R+r)(R-3r)$ as the sole necessary and sufficient condition for the
existence of a nested tetrahedron. Our method also gives the condition $d^2
\leq R(R-2r)$ for the existence of a nested triangle in the analogous
2-dimensional scenario. These results imply the Euler inequality in 2 and 3
dimensions. Furthermore, we formulate a new inequality that applies to the more
general case of ellipses and ellipsoids.
