1954 AHSME Problem 39

The locus of the midpoint of a line segment that is drawn from a given external point P to a given circle with center O and radius r, is:

\begin{array}{l} \textbf{(A)}\ \text{a straight line perpendicular to }\overline{PO}\\ \textbf{(B)}\ \text{a straight line parallel to }\overline{PO}\\ \textbf{(C)}\ \text{a circle with center }P\text{ and radius }r\\ \textbf{(D)}\ \text{a circle with center at the midpoint of }\overline{PO}\text{ and radius }2r\\ \textbf{(E)}\ \text{a circle with center at the midpoint }\overline{PO}\text{ and radius }\frac{1}{2}r \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.