Let
![A_1A_2A_3A_4](/media/m/9/f/c/9fc60bc7746a37e2c1a8fb688ba3a2ea.png)
be a tetrahedron,
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
its centroid, and
![A'_1, A'_2, A'_3,](/media/m/5/2/8/528d11234cc8889f97044205a93f417d.png)
and
![A'_4](/media/m/9/5/e/95e103f299fb9ccde5647219cbebf806.png)
the points where the circumsphere of
![A_1A_2A_3A_4](/media/m/9/f/c/9fc60bc7746a37e2c1a8fb688ba3a2ea.png)
intersects
![GA_1,GA_2,GA_3,](/media/m/0/5/c/05cc6a454139fa2c66f020e998b97541.png)
and
![GA_4,](/media/m/c/3/7/c37ee02be1a20fc097626bf4ea53cf65.png)
respectively. Prove that
and
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Let $A_1A_2A_3A_4$ be a tetrahedron, $G$ its centroid, and $A'_1, A'_2, A'_3,$ and $A'_4$ the points where the circumsphere of $A_1A_2A_3A_4$ intersects $GA_1,GA_2,GA_3,$ and $GA_4,$ respectively. Prove that
$$GA_1 \cdot GA_2 \cdot GA_3 \cdot GA_ \cdot4 \leq GA'_1 \cdot GA'_2 \cdot GA'_3 \cdot GA'_4$$
and
$$\frac{1}{GA'_1} + \frac{1}{GA'_2} + \frac{1}{GA'_3} + \frac{1}{GA'_4} \leq \frac{1}{GA_1} + \frac{1}{GA_2} + \frac{1}{GA_3} +...$$