1954 AMC 12 Problem 39

Let the variable endpoint on the given circle be $X$ and its midpoint with $P$ be $M$

The map $X\mapsto M$ is a dilation centered at $P$ with scale factor $\frac{1}{2}$

As the endpoint $X$ moves on the circle centered at $O,$ its midpoint $M$ with the fixed point $P$ is the image of $X$ under a dilation of scale $\frac{1}{2}$ centered at $P.$ Therefore the locus is a circle whose center is the midpoint of $\overline{PO}$ and whose radius is $\frac{r}{2}.$

Thus, the correct answer is E.