$20 \cdot 80$ In circle $O$, $G$ is a moving point on the diameter $AB$, construct $AA' \perp AB$ and $AA' = AG$, construct $BB' \perp AB$ and $BB' = BG$, $AA'$ and $BB'$ are on the same side of $AB$, $O'$ is the midpoint of $A'B'$, when $G$ moves from $A$ to $B$, then $O'$
(A) moves on a line bisecting $AB$.
(B) remains fixed.
(C) moves on a line perpendicular to $AB$.
(D) moves on a smaller circle intersecting the given circle.
(E) follows a path that is neither a circle nor a straight line.
(8th American High School Mathematics Examination, 1957)
[Solution] Connect $O O^{\prime}$. By the property of the midline of a trapezoid, we have
$$
\begin{aligned}
O O^{\prime} & =\frac{1}{2}\left(A A^{\prime}+B B^{\prime}\right) \\
& =\frac{1}{2}(A G+G B)=\frac{1}{2} A B,
\end{aligned}
$$

and $O O^{\prime} / / A A^{\prime}$.
$\therefore O O^{\prime} \perp A B$, so $O^{\prime}$ does not change with the movement of point $G$. Therefore, the answer is $(B)$.