$26 \cdot 35$ Let the left and right foci of a hyperbola be $F_{1}, F_{2}$, and the left and right vertices be $M, N$. If the vertex $P$ of $\triangle P F_{1} F_{2}$ is on the hyperbola, then the point where the incircle of $\triangle P F_{1} F_{2}$ touches the side $F_{1} F_{2}$ is
(A) inside the segment $M N$.
(B) inside the segment $F_{1} M$ or inside the segment $N F_{2}$.
(C) point $M$ or point $N$.
(D) cannot be determined.
(China High School Mathematics League, 1990)
[Solution] Let the incircle touch $F_{1} F_{2}$ at point $G$, touch $P F_{1}$ at $H$, and touch $P F_{2}$ at $K$. When $P$ is on the right branch, we get
$$
\begin{aligned}
& \left|G F_{1}\right|-\left|G F_{2}\right| \\
= & \left|H F_{1}\right|-\left|K F_{2}\right| \\
= & \left(\left|H F_{1}\right|+|H P|\right)-\left(\left|K F_{2}\right|+|K P|\right) \\
= & \left|P F_{1}\right|-\left|P F_{2}\right| .
\end{aligned}
$$

By the definition of a hyperbola, $G$ lies on the hyperbola, so $G$ coincides with $N$.
Similarly, if point $P$ is on the left branch, then $G$ coincides with $M$.
Therefore, the answer is $(C)$.