Let
be a triangle with centroid
. Determine, with proof, the position of the point
in the plane of
such that
is a minimum, and express this minimum value in terms of the side lengths of
.
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Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$.
Školjka