1969 AMC 12 Problem 10

Points equidistant from the two parallel tangents lie on their midway parallel line

On that line, compare distance to either tangent with distance to the circle

Let the circle have center $O$ and radius $r.$ A point equidistant from the two parallel tangents must lie on their midline, which passes through $O.$ If its distance from $O$ is $t,$ its distance from either tangent is $r,$ while its distance from the circle is $\lvert t-r\rvert.$ Thus $\lvert t-r\rvert=r,$ giving $t=0$ or $t=2r.$ There is one point with $t=0$ and two with $t=2r,$ for a total of $3.$

Therefore, the correct answer is C.