Y by Exponent11
Let 
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](//latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](//latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove that ![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
This post has been edited 19 times. Last edited by ReticulatedPython, Apr 22, 2025, 5:31 PM
,
,
Let the sphere that passes through these four points have equation 
See if you can find 
,
and 
.
