Trigonometry · Superchallenge problem

Triangle $ABC$ has side lengths $3$, $4$, and $5$. As $P$ ranges over all points of the plane, find the minimum value of $\overrightarrow{PA}\cdot\overrightarrow{PB} + \overrightarrow{PB}\cdot\overrightarrow{PC} + \overrightarrow{PC}\cdot\overrightarrow{PA}$.