For any triangle, the radius of its inscribed circle, the radius of its circumcircle and the distance of their centers are related through Euler's theorem in geometry (but earlier already published by Chapple). In one dimension higher, the Grace-Danielsson inequality gives a condition for the three values, so that a (non-regular) tetrahedron between the spheres exists, hence is completely contained inside the larger sphere and completely encloses the smaller sphere. In higher dimensions, Greg Egan conjectured a generalized Grace-Danielsson inequality and proved it to be sufficient for a simplex to exist between the spheres under a blog post of John Baez. A few weeks ago, the inequality was also proven to be necessary by Sergei Drozdov.
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