$25 \cdot 17$ Let $P$ be a point on the hypotenuse (or its extension) of an isosceles right triangle $A B C$. If $s=A P^{2}+B P^{2}$, then
(A) for a finite number of positions of $P$, $s=2 C P^{2}$.
(20th American High School Mathematics Examination, 1969)
[Solution] Establish a rectangular coordinate system as shown in the figure below, and let the coordinates of points $A, B$ be $(-a, 0),(a, 0)$, and the coordinates of point $P$ be $(x, 0)$. According to the problem,
$$
\begin{aligned}
s & =[x-(-a)]^{2}+(x-a)^{2} \\
& =2\left(x^{2}+a^{2}\right),
\end{aligned}
$$

Also, from the above setup, the coordinates of point $C$ are $(0, a)$. Therefore,
$$
C P^{2}=(0-x)^{2}+(a-0)^{2}=x^{2}+a^{2} \text {, }
$$

which means
$$
2 C P^{2}=s \text {. }
$$

Thus, the correct choice is $(D)$.