1. Let $P$ be a moving point in the plane of $\triangle A B C$. Then the point $P$ that minimizes $\boldsymbol{P A} \cdot \boldsymbol{P B}+\boldsymbol{P B} \cdot \boldsymbol{P C}+\boldsymbol{P C} \cdot \boldsymbol{P A}$ is the ( ) of $\triangle A B C$.
(A) Circumcenter
(B) Incenter
(C) Centroid
(D) Orthocenter
- 1.C.

Notice
$$
\begin{array}{l}
P A \cdot P B+P B \cdot P C+P C \cdot P A \\
=P A \cdot(P A+A B)+(P A+A B) \text {. } \\
(P A+A C)+(P A+A C) \cdot P A \\
=3 P A^{2}+2(A B+A C) \cdot P A+A B \cdot A C \\
=3\left(P A+\frac{A B+A C}{3}\right)^{2}-\frac{(A B+A C)^{2}}{3}+ \\
A B \cdot A C \text {. } \\
\end{array}
$$

When $P A=-\frac{A B+A C}{3}$, i.e., $P$ is the centroid of $\triangle A B C$, equation (1) achieves its minimum value
$$
-\frac{(A B+A C)^{2}}{3}+A B \cdot A C \text {. }
$$