$17 \cdot 173$ If a line bisects both the area and the perimeter of a triangle, then the line must pass through the
(A) incenter.
(B) circumcenter.
(C) centroid.
(D) orthocenter.
(China Junior High School Mathematics League, 1996)
[Solution] As shown in the figure, let the line that bisects the area and perimeter of $\triangle ABC$ intersect $AB$ and $AC$ at $D, E$, respectively, and let $I$ be the incenter of $\triangle ABC$, with $r$ being the radius of the incircle. Then we have
$$
\begin{array}{l}
S_{\text {IEAD }}=\frac{1}{2} r(A E+A D), \\
S_{B C E I D}=\frac{1}{2} r(B D+B C+C E) .
\end{array}
$$

Since line $ED$ bisects the perimeter of the triangle, we have
$$
A E+A D=B D+B C+C E,
$$

Thus, $\quad S_{\text {IEAD }}=S_{B C E I D}$.
Given that $S_{\triangle A D E}=S_{B C E D}$,
it follows that $S_{\triangle I D E}=0$, meaning line $DE$ passes through $I$.
Therefore, the correct choice is $(A)$.