Let
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
be a triangle with centroid
![G](/media/m/f/e/b/feb7f8fc95cee3c3a479382202e06a86.png)
. Determine, with proof, the position of the point
![P](/media/m/9/6/8/968d210d037e7e95372de185e8fb8759.png)
in the plane of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
such that
![AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG](/media/m/2/e/f/2ef6d24c42f58442f973f023fb5a2eff.png)
is a minimum, and express this minimum value in terms of the side lengths of
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
.
%V0
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 $ABC$.