13. Let $G$ be the centroid of $\triangle ABC$. Determine the position of point $P$ in the plane of $\triangle ABC$ such that $P A \cdot A G + B P \cdot B G + C P \cdot C G$ is minimized, and express this minimum value in terms of the side lengths of $\triangle ABC$.
(IMO - 42 Shortlist)
13. Let $a, b, c$ be the lengths of the sides opposite to vertices $A, B, C$ of $\triangle ABC$, respectively. We will prove that the minimum value is achieved when $P$ is the centroid $G$, and the minimum value is
$$
A G^{2}+B G^{2}+C G^{2}=\frac{1}{9}\left[\left(2 b^{2}+2 c^{2}-a^{2}\right)+\left(2 c^{2}+2 a^{2}-b^{2}\right)+\left(2 a^{2}+2 b^{2}-c^{2}\right)\right]=\frac{1}{3}\left(a^{2}+b^{2}+c^{2}\right) .
$$

Let $\Gamma$ be the circle passing through $B, G, C$, and the median $AL$ intersects $\Gamma$ at $G$ (inside $\triangle ABC$) and $K$. Let $\angle B G K=\theta, \angle A G N=\varphi$, $\angle B G N=\delta$ (where $N$ is the midpoint of $AB$). Since $L$ is the midpoint of $BC$, the distances from $B$ and $C$ to $GL$ are equal, i.e., $B G \cdot \sin \theta=C G \sin \varphi$, hence $\frac{B G}{\sin \varphi}=\frac{C G}{\sin \varphi}$. Similarly, $\frac{A G}{\sin \delta}=\frac{B G}{\sin \varphi}$. Let $R$ be the radius of circle $\Gamma$, then $B K=2 R \cdot \sin \theta, C K=2 R \cdot \sin \varphi, B C=2 R \cdot \sin \delta$, i.e., $\frac{A G}{B C}=\frac{B G}{C K}=\frac{C G}{B K} \cdot(*)$ Suppose $P$ is any point on the plane of $\triangle ABC$, by Ptolemy's inequality, we have $P K \cdot B C \leqslant B P \cdot C K + C P \cdot B K$, equality holds if and only if $P$ is on circle $\Gamma$. By (*) we have $P K \cdot A G \leqslant B P \cdot B G + C P \cdot C G$. Adding $A P \cdot A G$ to both sides, we get $(A P + P K) \cdot A G \leqslant A P \cdot A G + B P \cdot B G + G P \cdot C G$, and since $A K \leqslant A P + P K$, then $A K \cdot A G \leqslant A P \cdot A G + B P \cdot B G + C P \cdot C G$, equality holds if and only if $P$ is on segment $A K$ and $P$ is on circle $\Gamma$, i.e., equality holds if and only if $P$ coincides with $G$.