183. Sixteen-point sphere. Can the radius of the sixteen-point sphere of some tetrahedron be half the radius of the sphere circumscribed around this tetrahedron? (A sixteen-point sphere is called a sphere that passes through the centers of the circles circumscribed around the faces of the given tetrahedron.)
183. Let $R$ be the radius of the circumscribed sphere, and $r$ be the radius of the sixteen-point sphere. If the given tetrahedron is regular, then $r=\frac{R}{3}$; if, however, it has three right dihedral angles at some vertex, then $r=\infty$ (since in this case all the centers of the circumscribed circles lie in the same plane), while $R$ is finite. Since a regular tetrahedron can be continuously deformed into a tetrahedron with three right angles at a vertex, there exists an intermediate position in which $r=\frac{R}{2}$. Moreover, there are intermediate positions in which the ratio $\frac{r}{R}$ takes on any specified values greater than $\frac{1}{3}$.

[Howard Eves, A. M. M., 50, 389 (June 1943).]