$16 \cdot 11$ Let $P$ be a point inside circle $K$ but not coinciding with the center. Draw all chords of circle $K$ passing through $P$ and determine their midpoints. The locus of these midpoints is
(A) a circle with one point removed.
(B) if the distance from point $P$ to the center of circle $K$ is less than half the radius of circle $K$, it is a circle; otherwise, it is a circular arc less than $360^{\circ}$.
(C) a semicircle with one point removed.
(D) a semicircle.
(E) a circle.
(26th American High School Mathematics Examination, 1975)
[Solution] On the one hand, suppose $P$ is a known point, $O$ is the center of circle $K$, and $M$ is the midpoint of a certain chord $AB$ passing through point $P$. Since $\angle OMP=90^{\circ}$, $M$ lies on the circle with $OP$ as its diameter.

On the other hand, if $M$ is any point on the circle with $OP$ as its diameter, then the chord of the known circle passing through $P$ and $M$ (if $M=P$, the chord of the known circle is tangent to circle $OP$ at $P$) is perpendicular to $OM$. Therefore, $M$ is the midpoint of this chord.
Hence, the answer is $(E)$.