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$A D, B E$ and $C F$ are the medians of triangle $A B C$ whose centroid is $G$ . If the points $A, F, G$ and $E$ are concyclic, then

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$2 b^{2} = a^{2} + c^{2}$

$2 c^{2} = a^{2} + b^{2}$

$2 a^{2} = b^{2} + c^{2}$

$2 b^{2} = 2 a^{2} + c^{2}$

answer is C.

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Detailed Solution

$B G . B E = B F . B A$

$\Rightarrow \frac{2}{3} {(B E)}^{2} = \frac{1}{2} c^{2}$

$\Rightarrow \frac{2}{3} \times \frac{1}{4} (2 a^{2} + 2 c^{2} - b^{2}) = \frac{1}{2} c^{2}$

$\Rightarrow 2 a^{2} = b^{2} + c^{2}$

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