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Determine the tetrahedron $ABCD$ so that $\frac{L}{R}$ has the maximum value, where $L=AB+AC+AD+BC+BD+CD$, and $R$ is the radius of the circumsphere of $ABCD$. sorry about my bad English ^-^

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Can you find a formula for $R$ in terms of side lengths and angles? – Alex Becker Apr 3 '12 at 23:32
It seems "obvious" that the tetrahedron you want is regular. If so, formulas for the radius of the sphere are available on the web. – Ross Millikan Apr 4 '12 at 13:05
The regular tetrahedron has $L/R\doteq 9.8$, a flat square degenerate tetrahedron has $L/R\doteq9.65$. – Christian Blatter Feb 8 '13 at 10:37
@RossMillikan It is natural to think that the tetrahedron maximizing the given isoperimeter is probably regular. But do you have any clue how to prove it? – Gilles Bonnet Feb 14 '15 at 12:56

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