16.8 The number of points equidistant from a circle and two parallel tangents to the circle is
(A) 0.
(B) 2.
(C) 3.
(D) 4.
(E) infinitely many.
(20th American High School Mathematics Examination, 1969)
[Solution] Let $l_{1}, l_{2}$ be two parallel tangent lines of the known circle $\odot O$. If $r$ is the radius of $\odot O$, then the distance between $l_{1}, l_{2}$ is $2 r$.

The points equidistant from $l_{1}, l_{2}$ lie on the line $m$ (as shown in the figure), which is a line passing through $O$ and parallel to $l_{1}, l_{2}$.

The points at a distance of $r$ from the points on $\odot O$ lie on the circle with center $O$ and radius $2 r$, and it intersects $m$ at points $P, Q$.

In summary, points $P, Q, O$ satisfy the conditions of the problem.
Therefore, the answer is $(C)$.