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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} +...

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