1975 AHSME Problem 11

Let P be an interior point of circle K other than the center of K. Form all chords of K which pass through P, and determine their midpoints. The locus of these midpoints is

\begin{array}{l} \textbf{(A)}\ \text{a circle with one point deleted} \qquad \\ \textbf{(B)}\ \text{a circle if the distance from } P \text{ to the center of } K \text{ is less than} \\ \text{one half the radius of } K \text{; otherwise a circular arc of less than} 360^{\circ}\qquad \\ \textbf{(C)}\ \text{a semicircle with one point deleted} \qquad \\ \textbf{(D)}\ \text{a semicircle} \qquad \textbf{(E)}\ \text{a circle} \end{array}

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