2. If a straight line bisects both the area and the perimeter of a triangle, then the line must pass through the ( ) of the triangle.
(A) incenter
(B) circumcenter
(C) centroid
(D) orthocenter
(提示: Let line $D E$ bisect the perimeter and area of $\triangle A B C$, with $D 、 E$ on sides $A B 、 A C$ respectively. Draw the angle bisector of $\angle A$ intersecting $D E$ at point $P$, and let the distances from point $P$ to $A B 、 A C$ be $r$, and the distance from point $P$ to $B C$ be $r_{1}$. From
$$
\left\{\begin{array}{l}
A D+A E=D B+B C+C E, \\
\frac{r}{2}(A D+A E)=\frac{r}{2}(D B+C E)+\frac{r_{1}}{2} B C,
\end{array}\right.
$$

we solve to get $r_{1}=r$. Choose A.)